Chapter 14

Geophysical

1. Introduction ...... 603 11. with ...... 636 2. Vertical Variation of Density in Particle Orbit ...... 638 and ...... 605 Inertial ...... 639 3. ...... 607 12. Kelvin ...... 639 Formulation of the Frictional 13. Potential Conservation in Term...... 608 Shallow- Theory...... 644 4. Approximate Equations for a Thin 14. Internal Waves ...... 647 Layer on a Rotating Sphere...... 610 WKB Solution ...... 650 f -Plane Model ...... 612 Particle Orbit ...... 652 β-Plane Model ...... 612 Discussion of the 5. Geostrophic Flow...... 613 Relation...... 654 Thermal ...... 614 ...... 656 Taylor–Proudman Theorem ...... 615 15. ...... 657 6. at a ...... 617 Quasi-geostrophic Vorticity Explanation in Terms of Equation...... 658 Tilting ...... 622 ...... 660 7. Ekman Layer on a Rigid Surface ..... 622 16. Barotropic Instability...... 663 8. Shallow-Water Equations ...... 625 17. Baroclinic Instability ...... 665 9. Normal Modes in a Continuously Perturbation Vorticity Equation ...... 666 Stratified Layer ...... 628 Wave Solution ...... 668 Boundary Conditions on ψn ...... 630 Boundary Conditions ...... 669 Solution of Vertical Modes for Instability Criterion...... 669 Uniform N ...... 631 Energetics...... 671 10. High- and Low-Frequency Regimes in 18. Geostrophic ...... 673 Shallow-Water Equations...... 634 Exercises ...... 676 Literature Cited ...... 677

1. Introduction The subject of geophysical fluid dynamics deals with the dynamics of the atmosphere and the ocean. It has recently become an important branch of fluid dynamics due to our increasing interest in the environment. The field has been largely developed by meteorologists and oceanographers, but non-specialists have also been interested in the subject. Taylor was not a geophysical fluid dynamicist, but he held the position of

603 604 Geophysical

a meteorologist for some , and through this involvement he developed a special interest in the problems of turbulence and instability. Although Prandtl was mainly interested in the engineering aspects of fluid , his well-known textbook (Prandtl, 1952) contains several sections dealing with meteorological aspects of fluid mechanics. Notwithstanding the pressure for specialization that we all experience these days, it is worthwhile to learn something of this fascinating field even if one’s primary interest is in another of fluid mechanics. The importance of the study of atmospheric dynamics can hardly be overem- phasized. We live within the atmosphere and are almost helplessly affected by the and its rather chaotic behavior. The motion of the atmosphere is intimately connected with that of the ocean, with which it exchanges fluxes of , heat and moisture, and this makes the dynamics of the ocean as important as that of the atmosphere. The study of ocean currents is also important in its own right because of its relevance to navigation, fisheries, and disposal. The two features that distinguish geophysical fluid dynamics from other of fluid dynamics are the rotation of the and the vertical density stratification of the medium. We shall see that these two effects dominate the dynamics to such an extent that entirely new classes of phenomena arise, which have no counterpart in the laboratory scale flows we have studied in the preceding chapters. (For example, we shall see that the dominant mode of flow in the atmosphere and the ocean is along the lines of constant pressure, not from high to low pressures.) The motion of the atmosphere and the ocean is naturally studied in a coordinate frame rotating with the earth. This gives rise to the , which is discussed in Chapter 4. The density stratification gives rise to force, which is introduced in Chapter 4 (Conservation Laws) and discussed in further detail in Chapter 7 (Gravity Waves). In addition, important relevant material is discussed in Chapter 5 (Vorticity), Chapter 10 (), Chapter 12 (Instability), and Chapter 13 (Turbulence). The reader should be familiar with these before proceeding further with the present chapter. Because Coriolis and stratification effects play dominating roles in both the atmosphere and the ocean, there is a great deal of similarity between the dynam- ics of these two media; this makes it possible to study them together. There are also significant differences, however. For example the effects of lateral boundaries, due to the presence of continents, are important in the ocean but not in the atmosphere. The intense currents (like the and the Kuroshio) along the western boundaries of the ocean have no atmospheric analog. On the other hand phenomena like cloud formation and latent heat release due to moisture condensation are typically atmo- spheric phenomena. Processes are generally slower in the ocean, in which a typical horizontal is 0.1 m/s, although of the order of 1–2 m/s are found within the intense western boundary currents. In contrast, typical velocities in the atmosphere are 10–20 m/s. The nomenclature can also be different in the two fields. Meteorologists refer to a flow directed to the west as an “easterly wind” (i.e., from the east), while oceanographers refer to such a flow as a “westward current.” Atmospheric scientists refer to vertical positions by “heights” measured upward from the earth’s surface, while oceanographers refer to “depths” measured downward from the surface. However, we shall always take the vertical coordinate z to be upward, so no confusion should arise. 2. Vertical Variation of Density in Atmosphere and Ocean 605

We shall see that rotational effects caused by the presence of the have opposite signs in the two hemispheres. Note that all figures and descriptions given here are valid for the . In some cases the sense of the rotational effect for the has been explicitly mentioned. When the sense of the rotational effect is left unspecified for the southern hemisphere, it has to be assumed as opposite to that in the northern hemisphere.

2. Vertical Variation of Density in Atmosphere and Ocean An important variable in the study of geophysical fluid dynamics is the density strat- ification. In equation (1.38) we saw that the static stability of a fluid medium is determined by the sign of the potential density gradient

dρpot dρ gρ , (14.1) dz = dz + c2 where c is the of sound. A medium is statically stable if the potential density decreases with height. The first term on the right-hand side corresponds to the in situ density change due to all sources such as pressure, , and concentration of a constituent such as the in the sea or the water vapor in the atmosphere. The second term on the right-hand side is the density gradient due to the pressure decrease with height in an adiabatic environment and is called the adiabatic density gradient. The corresponding temperature gradient is called the adiabatic temperature gradient. For incompressible fluids c and the adiabatic density gradient is zero. As shown in Chapter 1, Section=∞ 10, the temperature of a dry adiabatic atmosphere decreases upward at the rate of 10 ◦C/km; that of a moist atmosphere decreases ≈ 2 at the rate of 5–6 ◦C/km. In the ocean, the adiabatic density gradient is gρ/c 3 ≈4 4 10− kg/m , taking a typical sonic speed of c 1520 m/s. The potential density ∼ × = 3 4 in the ocean increases with depth at a much smaller rate of 0.6 10− kg/m , so that the two terms on the right-hand side of equation (14.1) are× nearly in balance. It follows that most of the in situ density increase with depth in the ocean is due to the compressibility effects and not to changes in temperature or salinity. As potential density is the variable that determines the static stability, oceanographers take into account the compressibility effects by referring all their density measurements to the pressure. Unless specified otherwise, throughout the present chapter potential density will simply be referred to as “density,” omitting the qualifier “potential.” The mean vertical distribution of the in situ temperature in the lower 50 km of the atmosphere is shown in Figure 14.1. The lowest 10 km is called the , in which the temperature decreases with height at the rate of 6.5 ◦C/km. This is close to the moist adiabatic lapse rate, which means that the troposphere is close to being neutrally stable. The neutral stability is expected because turbulent mixing due to frictional and convective effects in the lower atmosphere keeps it well-stirred and therefore close to the neutral stratification. Practically all the clouds, weather changes, and water vapor of the atmosphere are found in the troposphere. The layer is capped by the , at an average height of 10 km, above which the temperature increases. This higher layer is called the , because it is very stably stratified. The 606 Geophysical Fluid Dynamics

Figure 14.1 Vertical distribution of temperature in the lower 50 km of the atmosphere.

increase of temperature with height in this layer is caused by the absorption of the ’s ultraviolet rays by ozone. The stability of the layer inhibits mixing and consequently acts as a lid on the turbulence and convective motion of the troposphere. The increase of temperature stops at the stratopause at a height of nearly 50 km. The vertical structure of density in the ocean is sketched in Figure 14.2, showing typical profiles of potential density and temperature. Most of the temperature increase with height is due to the absorption of solar radiation within the upper layer of the ocean. The density distribution in the ocean is also affected by the salinity. However, there is no characteristic variation of salinity with depth, and a decrease with depth is found to be as common as an increase with depth. In most cases, however, the vertical structure of density in the ocean is determined mainly by that of temperature, the salinity effects being secondary. The upper 50–200 m of ocean is well-mixed, due to the turbulence generated by the wind, waves, current shear, and the convective overturning caused by surface cooling. The temperature gradients decrease with depth, becoming quite small below a depth of 1500 m. There is usually a large temperature gradient in the depth range of 100–500 m. This layer of high stability is called the . Figure 14.2 also shows the profile of buoyancy frequency N, defined by g dρ N 2 , ≡−ρ0 dz

where ρ of course stands for the potential density and ρ0 is a constant reference density. 1 The buoyancy frequency reaches a typical maximum value of Nmax 0.01 s− (period 10 min) in the thermocline and decreases both upward and downward.∼ ∼ 3. Equations of Motion 607

Figure 14.2 Typical vertical distributions of: (a) temperature and density; and (b) buoyancy frequency in the ocean.

3. Equations of Motion In this section we shall review the relevant equations of motion, which are derived and discussed in Chapter 4. The equations of motion for a stratified medium, observed in a system of coordinates rotating at an ! with respect to the “fixed stars,” are

u 0, ∇ • = Du 1 gρ 2! u p k F, Dt + × =−ρ0 ∇ − ρ0 + (14.2) Dρ 0, Dt = where F is the force per unit . The diffusive effects in the density equation are omitted in set (14.2) because they will not be considered here. Set (14.2) makes the so-called Boussinesq approximation, discussed in Chapter 4, Section 18, in which the density variations are neglected everywhere except in the gravity term. Along with other restrictions, it assumes that the vertical scale of the motion is less than the “scale height” of the medium c2/g, where c is the speed of sound. This assumption is very good in the ocean, in which c2/g 200 km. In the atmosphere it is less applicable, because c2/g 10 km. Under the∼ Boussinesq ∼ approximation, the principle of mass conservation is expressed by • u 0. In contrast, the density equation Dρ/Dt 0 follows from the nondiffusive∇ heat equation= = DT /Dt 0 and an incompressible equation of state of the form δρ/ρ0 αδT . (If the density= is determined by the concentration S of a constituent, say= the− water vapor in the atmosphere or the salinity in the ocean, then Dρ/Dt 0 follows from = 608 Geophysical Fluid Dynamics

the nondiffusive conservation equation for the constituent in the form DS/Dt 0, = plus the incompressible equation of state δρ/ρ0 βδS.) The equations can be written in terms of the= pressure and density perturbations from a state of rest. In the absence of any motion, suppose the density and pressure have the vertical distributions ρ(z) and p(z), where the z-axis is taken vertically upward. As this state is hydrostatic,¯ we must¯ have dp ¯ ρg. (14.3) dz =−¯ In the presence of a flow field u(x,t), we can write the density and pressure as

ρ(x,t) ρ(z) ρ′(x, t), =¯ + (14.4) p(x,t) p(z) p′(x, t), =¯ +

where ρ′ and p′ are the changes from the state of rest. With this substitution, the first two terms on the right-hand side of the momentum equation in (14.2) give

1 gρ 1 g(ρ ρ′) p k (p p′) ¯ + k −ρ0 ∇ − ρ0 =−ρ0 ∇ ¯ + − ρ0

1 dp g(ρ ρ′) ¯ k p′ ¯ + k. =−ρ dz + ∇ − ρ 0 ! " 0 Subtracting the hydrostatic state (14.3), this becomes

1 gρ 1 gρ ′ p k p′ k, −ρ0 ∇ − ρ0 =−ρ0 ∇ − ρ0 which shows that we can replace p and ρ in equation (14.2) by the perturbation quantities p′ and ρ′.

Formulation of the Frictional Term The friction force per unit mass F in equation (14.2) needs to be related to the velocity field. From Chapter 4, Section 7, the friction force is given by

∂τij Fi , = ∂xj

where τij is the viscous tensor. The stress in a laminar flow is caused by the molecular exchanges of momentum. From equation (4.41), the viscous stress tensor in an isotropic incompressible medium in laminar flow is given by

∂ui ∂uj τij ρν . = ∂x + ∂x # j i $ In large-scale geophysical flows, however, the frictional forces are provided by turbu- lent mixing, and the molecular exchanges are negligible. The complexity of turbulent 3. Equations of Motion 609 behavior makes it impossible to relate the stress to the velocity field in a simple way. To proceed, then, we adopt the hypothesis, assuming that the turbulent stress is proportional to the velocity gradient field. Geophysical media are in the form of shallow stratified layers, in which the vertical velocities are much smaller than horizontal velocities. This means that the exchange of momentum across a horizontal surface is much weaker than that across a vertical surface. We expect then that the vertical eddy viscosity νv is much smaller than the horizontal eddy viscosity νH, and we assume that the turbulent stress components have the form ∂u ∂w τxz τzx ρνv ρνH , = = ∂z + ∂x ∂v ∂w τyz τzy ρνv ρνH , = = ∂z + ∂y (14.5) ∂u ∂v τxy τyx ρνH , = = ∂y + ∂x # $ ∂u ∂v ∂w τxx 2ρνH ,τyy 2ρνH ,τzz 2ρνv . = ∂x = ∂y = ∂z

The difficulty with set (14.5) is that the expressions for τxz and τyz depend on the fluid rotation in the vertical plane and not just the deformation. In Chapter 4, Section 10, we saw that a requirement for a constitutive equation is that the stresses should be independent of fluid rotation and should depend only on the deformation. There- fore, τxz should depend only on the combination (∂u/∂z ∂w/∂x), whereas the expression in equation (14.5) depends on both deformation+ and rotation. A tensori- ally correct geophysical treatment of the frictional terms is discussed, for example, in Kamenkovich (1967). However, the assumed form (14.5) leads to a simple formu- lation for viscous effects, as we shall see shortly. As the eddy viscosity assumption is of questionable validity (which Pedlosky (1971) describes as a “rather disreputable and desperate attempt”), there does not seem to be any purpose in formulating the stress–strain relation in more complicated ways merely to obey the requirement of invariance with respect to rotation. With the assumed form for the turbulent stress, the components of the frictional force Fi ∂τij /∂xj become = 2 2 2 ∂τxx ∂τxy ∂τxz ∂ u ∂ u ∂ u Fx νH νv , = ∂x + ∂y + ∂z = ∂x2 + ∂y2 + ∂z2 # $ 2 2 2 ∂τyx ∂τyy ∂τyz ∂ v ∂ v ∂ v Fy νH νv , (14.6) = ∂x + ∂y + ∂z = ∂x2 + ∂y2 + ∂z2 # $ 2 2 2 ∂τzx ∂τzy ∂τzz ∂ w ∂ w ∂ w Fz νH νv . = ∂x + ∂y + ∂z = ∂x2 + ∂y2 + ∂z2 # $ Estimates of the eddy coefficients vary greatly. Typical suggested values are 2 5 2 2 νv 10 m /s and νH 10 m /s for the lower atmosphere, and νv 0.01 m /s ∼ ∼ ∼ 610 Geophysical Fluid Dynamics

2 and νH 100 m /s for the upper ocean. In comparison, the molecular values are ν 1.5∼ 10 5 m2/s for air and ν 10 6 m2/s for water. = × − = − 4. Approximate Equations for a Thin Layer on a Rotating Sphere The atmosphere and the ocean are very thin layers in which the depth scale of flow is a few kilometers, whereas the horizontal scale is of the order of hundreds, or even thousands, of kilometers. The trajectories of fluid elements are very shallow and the vertical velocities are much smaller than the horizontal velocities. In fact, the suggests that the scale of the vertical velocity W is related to that of the horizontal velocity U by W H , U ∼ L where H is the depth scale and L is the horizontal length scale. Stratification and Coriolis effects usually constrain the vertical velocity to be even smaller than UH/L. Large-scale geophysical flow problems should be solved using spherical polar coordinates. If, however, the horizontal length scales are much smaller than the radius of the earth ( 6371 km), then the of the earth can be ignored, and the motion can be= studied by adopting a local Cartesian system on a tangent plane (Figure 14.3). On this plane we take an xyz , with x increasing eastward, y northward, and z upward. The corresponding velocity components are u (eastward), v (northward), and w (upward).

Figure 14.3 Local Cartesian coordinates. The x-axis is into the plane of the paper. 4. Approximate Equations for a Thin Layer on a Rotating Sphere 611

The earth rotates at a rate

4 1 ) 2π rad/day 0.73 10− s− , = = × around the polar axis, in a counterclockwise sense looking from above the north pole. From Figure 14.3, the components of angular velocity of the earth in the local Cartesian system are

)x 0, = )y ) cos θ, = )z ) sin θ, = where θ is the . The Coriolis force is therefore

ij k 2! u 02) cos θ 2) sin θ × = % % % uv w% % % % % 2% ) i(w cos θ v sin θ)% ju sin θ ku cos θ . = % [ − %+ − ] In the term multiplied by i we can use the condition w cos θ v sin θ, because the thin sheet approximation requires that w v. The three components≪ of the Coriolis force are therefore ≪

(2! u)x (2) sin θ)v fv, × =− =− (2! u)y (2) sin θ)u f u, (14.7) × = = (2! u)z (2) cos θ)u, × =− where we have defined

f 2) sin θ, (14.8) = to be twice the vertical component of !. As vorticity is twice the angular veloc- ity, f is called the planetary vorticity. More commonly, f is referred to as the Coriolis parameter, or the Coriolis frequency. It is positive in the northern hemi- 4 1 sphere and negative in the southern hemisphere, varying from 1.45 10− s− at the poles to zero at the . This makes sense, since a person± standing× at the north pole spins around himself in an counterclockwise sense at a rate ), whereas a person standing at the equator does not spin around himself but simply translates. The quantity

Ti 2π/f, = is called the inertial period, for reasons that will be clear in Section 11. 612 Geophysical Fluid Dynamics

The vertical component of the Coriolis force, namely 2)u cos θ, is generally negligible compared to the dominant terms in the vertical equation− of motion, namely 1 gρ ′/ρ0 and ρ0− (∂p′/∂z). Using equations (14.6) and (14.7), the equations of motion (14.2) reduce to

Du 1 ∂p ∂2u ∂2u ∂2u fv νH νv , Dt − =−ρ ∂x + ∂x2 + ∂y2 + ∂z2 0 # $ Dv 1 ∂p ∂2v ∂2v ∂2v fu νH νv , Dt + =−ρ ∂y + ∂x2 + ∂y2 + ∂z2 (14.9) 0 # $ Dw 1 ∂p gρ ∂2w ∂2w ∂2w νH νv . Dt =−ρ ∂z − ρ + ∂x2 + ∂y2 + ∂z2 0 0 # $

These are the equations of motion for a thin shell on a rotating earth. Note that only the vertical component of the earth’s angular velocity appears as a consequence of the flatness of the fluid trajectories.

f -Plane Model The Coriolis parameter f 2) sin θ varies with latitude θ. However, we shall see later that this variation is important= only for phenomena having very long time scales (several weeks) or very long length scales (thousands of kilometers). For many pur- poses we can assume f to be a constant, say f0 2) sin θ0, where θ0 is the central latitude of the region under study. A model using= a constant Coriolis parameter is called an f-plane model.

β-Plane Model The variation of f with latitude can be approximately represented by expanding f in a Taylor series about the central latitude θ0:

f f0 βy, (14.10) = + where we defined

df df dθ 2) cos θ β 0 . ≡ dy = dθ dy = R # $θ0 # $θ0 Here, we have used f 2) sin θ and dθ/dy 1/R, where the radius of the earth is nearly = =

R 6371 km. = A model that takes into account the variation of the Coriolis parameter in the simplified form f f0 βy, with β as constant, is called a β-plane model. = + 5. Geostrophic Flow 613

5. Geostrophic Flow Consider quasi-steady large-scale in the atmosphere or the ocean, away from boundaries. For these flows an excellent approximation for the horizontal equilibrium is a balance between the Coriolis force and the :

1 ∂p fv , − =−ρ0 ∂x (14.11) 1 ∂p fu . =−ρ0 ∂y

Here we have neglected the nonlinear terms, which are of order U 2/L, in comparison to the Coriolis force fU (U is the horizontal velocity scale, and L is the horizontal length scale.) The ratio∼ of the nonlinear term to the Coriolis term is called the :

Nonlinear acceleration U 2/L U Rossby number Ro. = Coriolis force ∼ fU = fL =

4 1 For a typical atmospheric value of U 10 m/s, f 10− s− , and L 1000 km, the Rossby number turns out to be 0.1.∼ The Rossby number∼ is even smaller∼ for many flows in the ocean, so that the neglect of nonlinear terms is justified for many flows. The balance of forces represented by equation (14.11), in which the horizontal pressure gradients are balanced by Coriolis forces, is called a geostrophic balance. In such a system the velocity distribution can be determined from a measured distribu- tion of the pressure field. The geostrophic equilibrium breaks down near the equator (within a latitude belt of 3◦), where f becomes small. It also breaks down if the frictional effects or unsteadiness± become important. Velocities in a geostrophic flow are perpendicular to the horizontal pressure gradient. This is because equation (14.11) implies that v p 0, i.e., . . . • ∇ = 1 ∂p ∂p ∂p ∂p (iu jv) p i j i j 0. + • ∇ = ρ f − ∂y + ∂x • ∂x + ∂y = 0 # $ # $ Thus, the horizontal velocity is along, and not across, the lines of constant pressure. If f is regarded as constant, then the geostrophic balance (14.11) shows that p/fρ0 can be regarded as a streamfunction. The isobars on a weather map are therefore nearly the streamlines of the flow. Figure 14.4 shows the geostrophic flow around low and high pressure centers in the northern hemisphere. Here the Coriolis force acts to the right of the velocity vector. This requires the flow to be counterclockwise (viewed from above) around a low pressure region and clockwise around a high pressure region. The sense of circulation is opposite in the southern hemisphere, where the Coriolis force acts to the left of the velocity vector. (Frictional forces become important at lower levels in the atmosphere and result in a flow partially across the isobars. This will be discussed 614 Geophysical Fluid Dynamics

Figure 14.4 Geostrophic flow around low and high pressure centers. The pressure force ( p) is −∇ indicated by a thin arrow, and the Coriolis force is indicated by a thick arrow.

in Section 7, where we will see that the flow around a low pressure center spirals inward due to frictional effects.) The flow along isobars at first surprises a reader unfamiliar with the effects of the Coriolis force. A question commonly asked is: How is such a motion set up? A typical manner of establishment of such a flow is as follows. Consider a horizontally converging flow in the of the ocean. The convergent flow sets up the sea surface in the form of a gentle “hill,” with the sea surface dropping away from the center of the hill. A fluid particle starting to move down the “hill” is deflected to the right in the northern hemisphere, and a steady state is reached when the particle finally moves along the isobars.

Thermal Wind In the presence of a horizontal gradient of density, the geostrophic velocity develops a vertical shear. This is easy to demonstrate from an analysis of the geostrophic and hydrostatic balance

1 ∂p fv , (14.12) − =−ρ0 ∂x 1 ∂p fu , (14.13) =−ρ0 ∂y ∂p 0 gρ. (14.14) =−∂z − 5. Geostrophic Flow 615

Eliminating p between equations (14.12) and (14.14), and also between equations (14.13) and (14.14), we obtain, respectively,

∂v g ∂ρ , ∂z =−ρ0f ∂x (14.15) ∂u g ∂ρ . ∂z = ρ0f ∂y

Meteorologists call these the equations because they give the verti- cal variation of wind from measurements of horizontal temperature gradients. The thermal wind is a baroclinic phenomenon, because the surfaces of constant p and ρ do not coincide.

Taylor–Proudman Theorem A striking phenomenon occurs in the geostrophic flow of a homogeneous fluid. It can only be observed in a laboratory experiment because stratification effects cannot be avoided in natural flows. Consider then a laboratory experiment in which a tank of fluid is steadily rotated at a high angular speed ) and a solid body is moved slowly along the bottom of the tank. The purpose of making ) large and the movement of the solid body slow is to make the Coriolis force much larger than the acceleration terms, which must be made negligible for geostrophic equilibrium. Away from the frictional effects of boundaries, the balance is therefore geostrophic in the horizontal and hydrostatic in the vertical: 1 ∂p 2)v , (14.16) − =−ρ ∂x 1 ∂p 2)u , (14.17) =−ρ ∂y 1 ∂p 0 g. (14.18) =−ρ ∂z − It is useful to define an as the ratio of viscous to Coriolis forces (per unit volume):

viscous force ρνU/L2 ν Ekman number E. = Coriolis force = ρf U = fL2 =

Under the circumstances already described here, both Ro and E are small. Elimination of p by cross differentiation between the horizontal momentum equations gives

∂v ∂u 2) 0. ∂y + ∂x = # $ 616 Geophysical Fluid Dynamics

Using the continuity equation, this gives

∂w 0. (14.19) ∂z =

Also, differentiating equations (14.16) and (14.17) with respect to z, and using equation (14.18), we obtain

∂v ∂u 0. (14.20) ∂z = ∂z =

Equations (14.19) and (14.20) show that

∂u 0, (14.21) ∂z =

showing that the velocity vector cannot vary in the direction of !. In other words, steady slow motions in a rotating, homogeneous, inviscid fluid are two dimensional. This is the Taylor–Proudman theorem, first derived by Proudman in 1916 and demon- strated experimentally by Taylor soon afterwards. In Taylor’s experiment, a tank was made to rotate as a solid body, and a small cylinder was slowly dragged along the bottom of the tank (Figure 14.5). Dye was introduced from point A above the cylinder and directly ahead of it. In a nonrotat- ing fluid the water would pass over the top of the moving cylinder. In the rotating experiment, however, the dye divides at a point S, as if it had been blocked by an upward extension of the cylinder, and flows around this imaginary cylinder, called the Taylor column. Dye released from a point B within the Taylor column remained there and moved with the cylinder. The conclusion was that the flow outside the upward extension of the cylinder is the same as if the cylinder extended across the entire water depth and that a column of water directly above the cylinder moves with it. The motion is two dimensional, although the solid body does not extend across the entire water depth. Taylor did a second experiment, in which he dragged a solid body parallel to the axis of rotation. In accordance with ∂w/∂z 0, he observed that a column of fluid is pushed ahead. The lateral velocity components= u and v were zero. In both of these experiments, there are shear layers at the edge of the Taylor column. In summary, Taylor’s experiment established the following striking fact for steady inviscid motion of homogeneous fluid in a strongly rotating system: Bodies moving either parallel or perpendicular to the axis of rotation carry along with their motion a so-called Taylor column of fluid, oriented parallel to the axis. The phenomenon is analogous to the horizontal blocking caused by a solid body (say a ) in a strongly stratified system, shown in Figure 7.33. 6. Ekman Layer at a Free Surface 617

Figure 14.5 Taylor’s experiment in a strongly rotating flow of a homogeneous fluid.

6. Ekman Layer at a Free Surface In the preceding section, we discussed a steady linear inviscid motion expected to be valid away from frictional boundary layers. We shall now examine the motion within frictional layers over horizontal surfaces. In viscous flows unaffected by Coriolis forces and pressure gradients, the only term which can balance the viscous force is either the time derivative ∂u/∂t or the u • u. The balance of ∂u/∂t and the viscous force gives rise to a viscous layer whose∇ thickness increases with time, as in the suddenly accelerated plate discussed in Chapter 9, Section 7. The balance of u • u and the viscous force give rise to a viscous layer whose thickness increases in the∇ direction of flow, as in the boundary layer over a semi-infinite plate discussed in Chapter 10, Sections 5 and 6. In a rotating flow, however, we can have a balance between the Coriolis and the viscous forces, and the thickness of the viscous layer can be invariant in time and . Two examples of such layers are given in this and the following sections. 618 Geophysical Fluid Dynamics

Consider first the case of a frictional layer near the free surface of the ocean, which is acted on by a τ in the x-direction. We shall not consider how the flow adjusts to the steady state but examine only the steady solution. We shall assume that the horizontal pressure gradients are zero and that the field is horizontally homogeneous. From equation (14.9), the horizontal equations of motion are

d2u fv νv , (14.22) − = dz2 d2v fu νv . (14.23) = dz2

Taking the z-axis vertically upward from the surface of the ocean, the boundary conditions are

du ρνv τ at z 0, (14.24) dz = = dv 0 at z 0, (14.25) dz = = u, v 0 as z . (14.26) → → −∞ Multiplying equation (14.23) by i √ 1 and adding equation (14.22), we obtain = − d2V if 2 V, (14.27) dz = νv where we have defined the “complex velocity”

V u iv. ≡ + The solution of equation (14.27) is

(1 i)z/δ (1 i)z/δ V Ae + Be− + , (14.28) = + where we have defined

2 ν δ v . (14.29) ≡ & f

We shall see shortly that δ is the thickness of the Ekman layer. The constant B is zero because the field must remain finite as z . The surface boundary → −∞ conditions (14.24) and (14.25) can be combined as ρνv(dV /dz) τ at z 0, from which equation (14.28) gives = = 6. Ekman Layer at a Free Surface 619

τδ(1 i) A − . = 2ρνv Substitution of this into equation (14.28) gives the velocity components

τ/ρ z π u ez/δ cos , = √fνv −δ + 4 ' ( τ/ρ z π v ez/δ sin . =−√fνv −δ + 4 ' ( The Swedish oceanographer Ekman worked out this solution in 1905. The solu- tion is shown in Figure 14.6 for the case of the northern hemisphere, in which f is positive. The velocities at various depths are plotted in Figure 14.6a, where each arrow represents the velocity vector at a certain depth. Such a plot of v vs u is some- called a “hodograph” plot. The vertical distributions of u and v are shown in Figure 14.6b. The hodograph shows that the surface velocity is deflected 45◦ to the right of the applied wind stress. (In the southern hemisphere the deflection is to the left of the surface stress.) The velocity vector rotates clockwise (looking down) with depth, and the magnitude exponentially decays with an e-folding scale of δ, which is called the Ekman layer thickness. The tips of the velocity vector at various depths form a spiral, called the .

Figure 14.6 Ekman layer at a free surface. The left panel shows velocity at various depths; values of z/δ are indicated along the curve traced out by the tip of the velocity vectors. The right panel shows − vertical distributions of u and v. 620 Geophysical Fluid Dynamics

The components of the volume transport in the Ekman layer are

0 udz 0, = )−∞ (14.30) 0 τ vdz . =−ρf )−∞ This shows that the net transport is to the right of the applied stress and is independent of νv. In fact, the result vdz τ/fρ follows directly from a vertical integration of the equation of motion in the= form− ρf v d(stress)/dz, so that the result does not depend on the eddy viscosity* assumption.− = The fact that the transport is to the right of the applied stress makes sense, because then the net (depth-integrated) Coriolis force, directed to the right of the depth-integrated transport, can balance the wind stress. The horizontal uniformity assumed in the solution is not a serious limitation. Since Ekman layers near the ocean surface have a thickness ( 50 m) much smaller than the scale of horizontal variation (L>100 km), the solution∼ is still locally appli- cable. The absence of horizontal pressure gradient assumed here can also be relaxed easily. Because of the thinness of the layer, any imposed horizontal pressure gradient remains constant across the layer. The presence of a horizontal pressure gradient merely adds a depth-independent geostrophic velocity to the Ekman solution. Suppose the sea surface slopes down to the north, so that there is a pressure force acting north- ward throughout the Ekman layer and below (Figure 14.7). This means that at the bottom of the Ekman layer (z/δ ) there is a geostrophic velocity U to the right of the pressure force. The surface→ −∞ Ekman spiral forced by the wind stress joins smoothly to this geostrophic velocity as z/δ . → −∞

Figure 14.7 Ekman layer at a free surface in the presence of a pressure gradient. The geostrophic velocity forced by the pressure gradient is U. 6. Ekman Layer at a Free Surface 621

Pure Ekman spirals are not observed in the surface layer of the ocean, mainly because the assumptions of constant eddy viscosity and steadiness are particularly restrictive. When the flow is averaged over a few days, however, several instances have been found in which the current does look like a spiral. One such example is shown in Figure 14.8.

Figure 14.8 An observed velocity distribution near the coast of Oregon. Velocity is averaged over 7 days. Wind stress had a magnitude of 1.1 dyn/cm2 and was directed nearly southward, as indicated at the top of the figure. The upper panel shows vertical distributions of u and v, and the lower panel shows the hodograph in which depths are indicated in meters. The hodograph is similar to that of a surface Ekman layer (of depth 16 m) lying over the bottom Ekman layer (extending from a depth of 16 m to the ocean bottom). P. Kundu, in Bottom Tubulence, J. C. J. Nihoul, ed., Elsevier, 1977 and reprinted with the permission of Jacques C. J. Nihoul. 622 Geophysical Fluid Dynamics

Explanation in Terms of Vortex Tilting We have seen in previous chapters that the thickness of a viscous layer usually grows in a nonrotating flow, either in time or in the direction of flow. The Ekman solution, in contrast, results in a viscous layer that does not grow either in time or space. This can be explained by examining the vorticity equation (Pedlosky, 1987). The vorticity components in the x- and y-directions are

∂w ∂v dv ωx , = ∂y − ∂z =−dz ∂u ∂w du ωy , = ∂z − ∂x = dz

where we have used w 0. Using these, the z-derivative of the equations of motion (14.22) and (14.23) gives=

2 dv d ωy f νv , − dz = dz2 (14.31) 2 du d ωx f νv . − dz = dz2 The right-hand side of these equations represent diffusion of vorticity. Without Coriolis forces this diffusion would cause a thickening of the viscous layer. The presence of planetary rotation, however, means that vertical fluid lines coincide with the planetary vortex lines. The tilting of vertical fluid lines, represented by terms on the left-hand sides of equations (14.31), then causes a rate of change of horizontal component of vorticity that just cancels the diffusion term.

7. Ekman Layer on a Rigid Surface Consider now a horizontally independent and steady viscous layer on a solid surface in a rotating flow. This can be the atmospheric boundary layer over the solid earth or the boundary layer over the ocean bottom. We assume that at large distances from the surface the velocity is toward the x-direction and has a magnitude U. Viscous forces are negligible far from the wall, so that the Coriolis force can be balanced only by a pressure gradient:

1 dp fU . (14.32) =−ρ dy

This simply states that the flow outside the viscous layer is in geostrophic balance, U being the geostrophic velocity. For our assumed case of positive U and f , we must have dp/dy < 0, so that the pressure falls with y—that is, the pressure force is directed along the positive y direction, resulting in a geostrophic flow U to the right 7. Ekman Layer on a Rigid Surface 623 of the pressure force in the northern hemisphere. The horizontal pressure gradient remains constant within the thin boundary layer. Near the solid surface the viscous forces are important, so that the balance within the boundary layer is

d2u fv νv , (14.33) − = dz2 d2v fu νv fU, (14.34) = dz2 +

1 where we have replaced ρ− (dp/dy) by fU in accordance with equation (14.32). The boundary conditions− are

u U, v 0 as z , (14.35) = = →∞ u 0,v0 at z 0, (14.36) = = = where z is taken vertically upward from the solid surface. Multiplying equation (14.34) by i and adding equation (14.33), the equations of motion become

d2V if 2 (V U), (14.37) dz = νv − where we have defined the complex velocity V u iv. The boundary conditions (14.35) and (14.36) in terms of the complex≡ velocity+ are

V U as z , (14.38) = →∞ V 0 at z 0. (14.39) = = The particular solution of equation (14.37) is V U. The total solution is, therefore, = (1 i)z/δ (1 i)z/δ V Ae− + Be + U, (14.40) = + + where δ √2νv/f . To satisfy equation (14.38), we must have B 0. Condition (14.39) gives≡ A U. The velocity components then become = =− z/δ u U 1 e− cos (z/δ) , = [ − ] (14.41) v Ue z/δ sin (z/δ). = − According to equation (14.41), the tip of the velocity vector describes a spiral for various values of z (Figure 14.9a). As with the Ekman layer at a free surface, the frictional effects are confined within a layer of thickness δ √2νv/f , which increases = with νv and decreases with the rotation rate f . Interestingly, the layer thickness is independent of the magnitude of the free-stream velocity U; this behavior is quite different from that of a steady nonrotating boundary layer on a semi-infinite plate (the Blasius solution of Section 10.5) in which the thickness is proportional to 1/√U. 624 Geophysical Fluid Dynamics

Figure 14.9 Ekman layer at a rigid surface. The left panel shows velocity vectors at various heights; values of z/δ are indicated along the curve traced out by the tip of the velocity vectors. The right panel shows vertical distributions of u and v.

Figure 14.9b shows the vertical distribution of the velocity components. Far from the wall the velocity is entirely in the x-direction, and the Coriolis force balances the pressure gradient. As the wall is approached, retarding effects decrease u and the associated Coriolis force, so that the pressure gradient (which is independent of z) forces a component v in the direction of the pressure force. Using equation (14.41), the net transport in the Ekman layer normal to the uniform stream outside the layer is

ν 1/2 1 ∞ vdz U v Uδ, = 2f = 2 )0 ! " which is directed to the left of the free-stream velocity, in the direction of the pressure force. If the atmosphere were in laminar motion, νv would be equal to its molecular 4 1 value for air, and the Ekman layer thickness at a latitude of 45◦ (where f 10− s− ) would be δ 0.4 m. The observed thickness of the atmospheric boundary≃ layer ≈ ∼ 2 is of order 1 km, which implies an eddy viscosity of order νv 50 m /s. In fact, Taylor (1915) tried to estimate the eddy viscosity by matching the∼ predicted velocity distributions (14.41) with the observed wind at various heights. The Ekman layer solution on a solid surface demonstrates that the three-way balance among the Coriolis force, the pressure force, and the frictional force within the boundary layer results in a component of flow directed toward the lower pressure. 8. Shallow-Water Equations 625

Figure 14.10 Balance of forces within an Ekman layer, showing that velocity u has a component toward low pressure.

The balance of forces within the boundary layer is illustrated in Figure 14.10. The net frictional force on an element is oriented approximately opposite to the velocity vector u. It is clear that a balance of forces is possible only if the velocity vector has a component from high to low pressure, as shown. Frictional forces therefore cause the flow around a low-pressure center to spiral inward. Mass conservation requires that the inward converging flow should rise over a low-, resulting in cloud formation and rainfall. This is what happens in a , which is a low-pressure system. In contrast, over a high-pressure system the air sinks as it spirals outward due to frictional effects. The arrival of high-pressure systems therefore brings in clear skies and fair weather, because the sinking air does not result in cloud formation. Frictional effects, in particular the by surface , play a fundamental role in the theory of wind-driven ocean circulation. Possibly the most important result of such theories was given by in 1948. He showed that the northward increase of the Coriolis parameter f is responsible for making the currents along the western boundary of the ocean (e.g., the Gulf Stream in the Atlantic and the Kuroshio in the Pacific) much stronger than the currents on the eastern side. These are discussed in books on physical and will not be presented here. Instead, we shall now turn our attention to the influence of Coriolis forces on inviscid wave motions.

8. Shallow-Water Equations Both surface and internal gravity waves were discussed in Chapter 7. The effect of planetary rotation was assumed to be small, which is valid if the frequency ω of the wave is much larger than the Coriolis parameter f . In this chapter we are considering phenomena slow enough for ω to be comparable to f . Consider surface 626 Geophysical Fluid Dynamics

gravity waves on a shallow layer of homogeneous fluid whose mean depth is H. If we restrict ourselves to wavelengths λ much larger than H , then the vertical velocities are much smaller than the horizontal velocities. In Chapter 7, Section 6 we saw that the acceleration ∂w/∂t is then negligible in the vertical momentum equation, so that the pressure distribution is hydrostatic. We also demonstrated that the fluid particles execute a horizontal rectilinear motion that is independent of z. When the effects of planetary rotation are included, the horizontal velocity is still depth-independent, although the particle orbits are no longer rectilinear but elliptic on a horizontal plane, as we shall see in the following section. Consider a layer of fluid over a flat horizontal bottom (Figure 14.11). Let z be measured upward from the bottom surface, and η be the of the free surface. The pressure at height z from the bottom, which is hydrostatic, is given by

p ρg(H η z). = + − The horizontal pressure gradients are therefore ∂p ∂η ∂p ∂η ρg , ρg . (14.42) ∂x = ∂x ∂y = ∂y As these are independent of z, the resulting horizontal motion is also depth independent. Now consider the continuity equation ∂u ∂v ∂w 0. ∂x + ∂y + ∂z = As ∂u/∂x and ∂v/∂y are independent of z, the continuity equation requires that w vary linearly with z, from zero at the bottom to the maximum value at the free surface. Integrating vertically across the from z 0 to z H η, and noting that u and v are depth independent, we obtain = = + ∂u ∂v (H η) (H η) w(η) w(0) 0, (14.43) + ∂x + + ∂y + − =

Figure 14.11 Layer of fluid on a flat bottom. 8. Shallow-Water Equations 627 where w(η) is the vertical velocity at the surface and w(0) 0 is the vertical velocity at the bottom. The surface velocity is given by =

Dη ∂η ∂η ∂η w(η) u v . = Dt = ∂t + ∂x + ∂y

The continuity equation (14.43) then becomes

∂u ∂v ∂η ∂η ∂η (H η) (H η) u v 0, + ∂x + + ∂y + ∂t + ∂x + ∂y = which can be written as

∂η ∂ ∂ u(H η) v(H η) 0. (14.44) ∂t + ∂x[ + ]+∂y [ + ]=

This says simply that the divergence of the horizontal transport depresses the free surface. For small amplitude waves, the quadratic nonlinear terms can be neglected in comparison to the linear terms, so that the divergence term in equation (14.44) simplifies to H • u. The linearized∇ continuity and momentum equations are then

∂η ∂u ∂v H 0, ∂t + ∂x + ∂y = # $ ∂u ∂η fv g , (14.45) ∂t − =− ∂x ∂v ∂η fu g . ∂t + =− ∂y

In the momentum equations of (14.45), the pressure gradient terms are written in the form (14.42) and the nonlinear advective terms have been neglected under the small amplitude assumption. Equations (14.45), called the , govern the motion of a layer of fluid in which the horizontal scale is much larger than the depth of the layer. These equations will be used in the following sections for studying various types of gravity waves. Although the preceding analysis has been formulated for a layer of homogeneous fluid, equations (14.45) are applicable to internal waves in a stratified medium, if we replaced H by the equivalent depth He, defined by

2 c gHe, (14.46) = where c is the speed of long nonrotating internal gravity waves. This will be demon- strated in the following section. 628 Geophysical Fluid Dynamics

9. Normal Modes in a Continuously Stratified Layer In the preceding section we considered a homogeneous medium and derived the governing equations for waves of wavelength larger than the depth of the fluid layer. Now consider a continuously stratified medium and assume that the horizontal scale of motion is much larger than the vertical scale. The pressure distribution is therefore hydrostatic, and the equations of motion are

∂u ∂v ∂w 0, (14.47) ∂x + ∂y + ∂z = ∂u 1 ∂p fv , (14.48) ∂t − =−ρ0 ∂x ∂v 1 ∂p fu , (14.49) ∂t + =−ρ0 ∂y ∂p 0 gρ, (14.50) =−∂z − ∂ρ ρ N 2 0 w 0, (14.51) ∂t − g =

where p and ρ represent perturbations of pressure and density from the state of rest. The advective term in the density equation is written in the linearized form 2 w(dρ/dz) ρ0N w/g, where N(z) is the buoyancy frequency. In this form the rate of¯ change=− of density at a point is assumed to be due only to the vertical advection of the background density distribution ρ(z), as discussed in Chapter 7, Section 18. In a continuously stratified medium,¯ it is convenient to use the method of separa- tion of variables and write q qn(x, y, t)ψn(z) for some variable q. The solution is thus written as the sum of various= vertical “modes,” which are called normal modes because they turn out to be orthogonal+ to each other. The vertical structure of a mode is described by ψn and qn describes the horizontal propagation of the mode. Although each mode propagates only horizontally, the sum of a number of modes can also propagate vertically if the various qn are out of phase. We assume separable solutions of the form

∞ u, v, p/ρ0 un,vn,pn ψn(z), (14.52) [ ]= [ ] n 0 ,= ∞ z w wn ψn(z) dz, (14.53) = n 0 ) H ,= − ∞ dψn ρ ρn , (14.54) = dz n 0 ,= 9. Normal Modes in a Continuously Stratified Layer 629

where the amplitudes un, vn, pn, wn, and ρn are functions of (x, y, t). The z-axis is measured from the upper free surface of the fluid layer, and z H represents the bottom wall. The reasons for assuming the various forms of=z-dependence− in equations (14.52)–(14.54) are the following: Variables u, v, and p have the same vertical structure in order to be consistent with equations (14.48) and (14.49). Con- tinuity equation (14.47) requires that the vertical structure of w should be the inte- gral of ψn(z). Equation (14.50) requires that the vertical structure of ρ must be the z-derivative of the vertical structure of p. Subsititution of equations (14.53) and (14.54) into equation (14.51) gives

2 z ∞ ∂ρn dψn ρ0N wn ψn dz 0. ∂t dz − g = n 0 ! ) H " ,= − This is valid for all values of z, and the modes are linearly independent, so the quantity within [ ] must vanish for each mode. This gives

dψ /dz ρ w 1 n 0 n . (14.55) 2 z 2 N H ψn dz = g ∂ρn/∂t ≡−cn − * As the first term is a function of z alone and the second term is a function of (x, y, t) alone, for consistency both terms must be equal to a constant; we take the “separation constant” to be 1/c2. The vertical structure is then given by − n z 1 dψn 1 2 2 ψn dz. N dz =−cn H )− Taking the z-derivative,

d 1 dψn 1 ψn 0, (14.56) dz N 2 dz + c2 = # $ n which is the differential equation governing the vertical structure of the normal modes. Equation (14.56) has the so-called Sturm–Liouville form, for which the various solu- tions are orthogonal. Equation (14.55) also gives

g ∂ρn wn 2 . =−ρ0cn ∂t Substitution of equations (14.52)–(14.54) into equations (14.47)–(14.51) finally gives the normal mode equations

∂un ∂vn 1 ∂pn 2 0, (14.57) ∂x + ∂y + cn ∂t = 630 Geophysical Fluid Dynamics

∂un ∂pn fvn , (14.58) ∂t − =−∂x

∂vn ∂pn fun , (14.59) ∂t + =−∂y g pn ρn, (14.60) =−ρ0

1 ∂pn wn 2 . (14.61) = cn ∂t

Once equations (14.57)–(14.59) have been solved for un, vn and pn, the amplitudes ρn and wn can be obtained from equations (14.60) and (14.61). The set (14.57)–(14.59) is identical to the set (14.45) governing the motion of a homogeneous layer, provided 2 pn is identified with gη and cn is identified with gH. In a stratified flow each mode (having a fixed vertical structure) behaves, in the horizontal dimensions and in time, just like a homogeneous layer, with an equivalent depth He defined by

2 c gHe. (14.62) n ≡

Boundary Conditions on ψn At the layer bottom, the boundary condition is

w 0 at z H. = =−

To write this condition in terms of ψn, we first combine the hydrostatic equation (14.50) and the density equation (14.51) to give w in terms of p:

g(∂ρ/∂t) 1 ∂2p 1 ∞ ∂p dψ w n n . (14.63) = ρ N 2 =−ρ N 2 ∂z∂t =−N 2 ∂t dz 0 0 n 0 ,= The requirement w 0 then yields the bottom boundary condition = dψ n 0 at z H. (14.64) dz = =−

We now formulate the surface boundary condition. The linearized surface boundary conditions are ∂η w ,p ρ0gη at z 0, (14.65 ) = ∂t = = ′ 9. Normal Modes in a Continuously Stratified Layer 631 where η is the free surface displacement. These conditions can be combined into ∂p ρ0gw at z 0. ∂t = = Using equation (14.63) this becomes

g ∂2p ∂p 0 at z 0. N 2 ∂z∂t + ∂t = = Substitution of the normal mode decomposition (14.52) gives

2 dψn N ψn 0 at z 0. (14.65) dz + g = =

The boundary conditions on ψn are therefore equations (14.64) and (14.65).

Solution of Vertical Modes for Uniform N

For a medium of uniform N, a simple solution can be found for ψn. From equa- tions (14.56), (14.64), and (14.65), the vertical structure of the normal modes is given by 2 2 d ψn N 2 2 ψn 0, (14.66) dz + cn = with the boundary conditions

2 dψn N ψn 0 at z 0, (14.67) dz + g = = dψ n 0 at z H. (14.68) dz = =−

The set (14.66)–(14.68) defines an eigenvalue problem, with ψn as the eigenfunction and cn as the eigenvalue. The solution of equation (14.66) is Nz Nz ψn An cos Bn sin . (14.69) = cn + cn Application of the surface boundary condition (14.67) gives

cnN Bn An. =− g The bottom boundary condition (14.68) then gives NH c N tan n , (14.70) cn = g whose roots define the eigenvalues of the problem. 632 Geophysical Fluid Dynamics

Figure 14.12 Calculation of eigenvalues cn of vertical normal modes in a fluid layer of depth H and uniform stratification N.

The solution of equation (14.70) is indicated graphically in Figure 14.12. The first root occurs for NH/cn 1, for which we can write tan(NH/cn) NH/cn, so that equation (14.70) gives≪ (indicating this root by n 0) ≃ =

c0 gH. = The vertical modal structure is found from- equation (14.69). Because the magnitude of an eigenfunction is arbitrary, we can set A0 1, obtaining = 2 Nz c0N Nz N z ψ0 cos sin 1 1, = c0 − g c0 ≃ − g ≃

2 where we have used N z /c0 1 (with NH/c0 1), and N z/g 1 (with 2 | | ≪ ≪ ≪ N H/g (NH/c0)(c0N/g) 1, both sides of equation (14.70) being much less than 1).= For this mode the vertical≪ structure of u, v, and p is therefore nearly depth-independent. The corresponding structure for w (given by ψ0 dz, as indi- cated in equation (14.53)) is linear in z, with zero at the bottom and a maximum at the upper free surface. A stratified medium therefore has a mode of motion* that behaves like that in an unstratified medium; this mode does not feel the stratification. The n 0 mode is called the barotropic mode. = The remaining modes n 1 are baroclinic. For these modes cnN/g 1 but ! ≪ NH/cn is not small, as can be seen in Figure 14.12, so that the baroclinic roots of equation (14.70) are nearly given by NH tan 0, cn = 9. Normal Modes in a Continuously Stratified Layer 633 which gives

NH cn , n 1, 2, 3,.... (14.71) = nπ =

Taking a typical depth-average oceanic value of N 10 3 s 1 and H 5 km, the ∼ − − ∼ eigenvalue for the first baroclinic mode is c1 2m/s. The corresponding equivalent 2 ∼ depth is He c1/g 0.4 m. An examination= ∼ of the algebraic steps leading to equation (14.70) shows that neglecting the right-hand side is equivalent to replacing the upper boundary condi- tion (14.65′) by w 0 at z 0. This is called the rigid lid approximation. The baroclinic modes are= negligibly= distorted by the rigid lid approximation. In contrast, the rigid lid approximation applied to the barotropic mode would yield c0 , as equation (14.71) shows for n 0. Note that the rigid lid approximation= does∞not imply that the free surface displacement= corresponding to the baroclinic modes is negligible in the ocean. In fact, excluding the wind waves and , much of the free surface displacements in the ocean are due to baroclinic motions. The rigid lid approximation merely implies that, for baroclinic motions, the vertical displacements at the surface are much smaller than those within the fluid column. A valid baroclinic solution can therefore be obtained by setting w 0 at z 0. Further, the rigid lid approximation does not imply that the pressure is= constant= at the level surface z 0; if a rigid lid were actually imposed at z 0, then the pressure on the lid would= vary due to the baroclinic motions. = The vertical mode shape under the rigid lid approximation is given by the cosine distribution nπz ψn cos ,n 0, 1, 2,..., = H = because it satisfies dψn/dz 0 at z 0, H . The nth mode ψn has n zero crossings within the layer (Figure 14.13).= = − A decomposition into normal modes is only possible in the absence of topographic variations and mean currents with shear. It is valid with or without Cori- olis forces and with or without the β-effect. However, the hydrostatic approximation here means that the frequencies are much smaller than N. Under this condition the eigenfunctions are independent of the frequency, as equation (14.56) shows. With- out the hydrostatic approximation the eigenfunctions ψn become dependent on the frequency ω. This is discussed, for example, in LeBlond and Mysak (1978). Summary: Small amplitude motion in a frictionless continuously stratified ocean can be decomposed in terms of noninteracting vertical normal modes. The vertical structure of each mode is defined by an eigenfunction ψn(z). If the horizontal scale of the waves is much larger than the vertical scale, then the equations governing the horizontal propagation of each mode are identical to those of a shallow homo- geneous layer, with the layer depth H replaced by an equivalent depth He defined 2 by c gHe. For a medium of constant N, the baroclinic (n 1) eigenvalues are n = ! 634 Geophysical Fluid Dynamics

Figure 14.13 Vertical distribution of a few normal modes in a stratified medium of uniform buoyancy frequency.

given by cn NH/πn, while the barotropic eigenvalue is c0 √gH. The rigid lid approximation= is quite good for the baroclinic modes. =

10. High- and Low-Frequency Regimes in Shallow-Water Equations We shall now examine what terms are negligible in the shallow-water equations for the various frequency ranges. Our analysis is valid for a single homogeneous layer or for a stratified medium. In the latter case H has to be interpreted as the equivalent depth, and c has to be interpreted as the speed of long nonrotating internal gravity waves. The β-effect will be considered in this section. As f varies only northward, horizontal isotropy is lost whenever the β-effect is included, and it becomes necessary to distinguish between the different horizontal directions. We shall follow the usual geophysical convention that the x-axis is directed eastward and the y-axis is directed northward, with u and v the corresponding velocity components. The simplest way to perform the analysis is to examine the v-equation. A single equation for v can be derived by first taking the time derivatives of the momentum equations in (14.45) and using the continuity equation to eliminate ∂η/∂t. This gives ∂2u ∂v ∂ ∂u ∂v f gH , (14.72) ∂t2 − ∂t = ∂x ∂x + ∂y # $ ∂2v ∂u ∂ ∂u ∂v f gH . (14.73) ∂t2 + ∂t = ∂y ∂x + ∂y # $ 10. High- and Low-Frequency Regimes in Shallow-Water Equations 635

Now take ∂/∂t of equation (14.73) and use equation (14.72), to obtain

∂3v ∂v ∂ ∂u ∂v ∂2 ∂u ∂v f f gH gH . (14.74) ∂t3 + ∂t + ∂x ∂x + ∂y = ∂y∂t ∂x + ∂y ! # $" # $ To eliminate u, we first obtain a vorticity equation by cross differentiating and sub- tracting the momentum equations in equation (14.45):

∂ ∂u ∂v ∂u ∂v f0 βv 0. ∂t ∂y − ∂x − ∂x + ∂y − = # $ # $ Here, we have made the customary β-plane approximation, valid if the y-scale is small enough so that /f /f 1. Accordingly, we have treated f as constant (and replaced ≪ it by an average value f0) except when df/dy appears; this is why we have written f0 in the second term of the preceding equation. Taking the x-derivative, multiplying by gH, and adding to equation (14.74), we finally obtain a vorticity equation in terms of v only:

∂3v ∂ ∂v ∂v gH 2 v f 2 gHβ 0, ∂t3 − ∂t∇H + 0 ∂t − ∂x = (14.75)

2 2 2 2 2 where H ∂ /∂x ∂ /∂y is the horizontal Laplacian operator. Equation∇ = (14.75)+ is Boussinesq, linear and hydrostatic, but otherwise quite gen- eral in the sense that it is applicable to both high and low frequencies. Consider wave solutions of the form

i(kx ly ωt) v ve + − , =ˆ where k is the eastward and l is the northward wavenumber. Then equa- tion (14.75) gives

ω3 c2ωK2 f 2ω c2βk 0, (14.76) − − 0 − = where K2 k2 l2 and c √gH. It can be shown that all roots of equation (14.76) are real, two= of the+ roots being= superinertial (ω>f) and the third being subinertial (ω f ). Equation (14.76) is the complete dispersion relation for linear shallow-water equations.≪ In various parametric ranges it takes simpler forms, representing simpler waves. First, consider high-frequency waves ω f . Then the third term of equa- tion (14.76) is negligible compared to the first≫ term. Moreover, the fourth term is also negligible in this range. Compare, for example, the fourth and second terms:

2 c βk β 3 10− , c2ωK2 ∼ ωK ∼ 636 Geophysical Fluid Dynamics

11 1 1 where we have assumed typical values of β 2 10− m− s− , ω 3f 4 1 = × = 3 10− s− , and 2π/K 100 km. For ω f , therefore, the balance is between the∼ first× and second terms in equation∼ (14.76), and≫ the roots are ω K√gH, which correspond to a propagation speed of ω/K √gH. The effects=± of both f and β are therefore negligible for high-frequency waves,= as is expected as they are too fast to be affected by the Coriolis effects. Next consider ω>f, but ω f . Then the third term in equation (14.76) is not negligible, but the β-effect is. These∼ are gravity waves influenced by Coriolis forces; gravity waves are discussed in the next section. However, the time scales are still too short for the motion to be affected by the β-effect. Last, consider very slow waves for which ω f . Then the β-effect becomes important, and the first term in equation (14.76) becomes≪ negligible. Compare, for example, the first and the last terms:

ω3 1. c2βk ≪

Typical values for the ocean are c 200 m/s for the barotropic mode, c 2m/s for ∼11 1 1 ∼ 5 1 the baroclinic mode, β 2 10− m− s− ,2π/k 100 km, and ω 10− s− . = × ∼4 ∼ This makes the forementioned ratio about 0.2 10− for the barotropic mode and 0.2 for the baroclinic mode. The first term in equation× (14.76) is therefore negligible for ω f . Equation≪ (14.75) governs the dynamics of a variety of wave motions in the ocean and the atmosphere, and the discussion in this section shows what terms can be dropped under various limiting conditions. An understanding of these limiting conditions will be useful in the following sections.

11. Gravity Waves with Rotation In this chapter we shall examine several free-wave solutions of the shallow-water equations. In this section we shall study gravity waves with frequencies in the range ω>f, for which the β-effect is negligible, as demonstrated in the preced- ing section. Consequently, the Coriolis frequency f is regarded as constant here. Consider progressive waves of the form

i(kx ly ωt) (u, v, η) (u, v,η)e + − , = ˆ ˆ ˆ where u, v, and η are the complex amplitudes, and the real part of the right-hand side is meant.ˆ ˆ Then equationˆ (14.45) gives

iωu f v ikgη, (14.77) − ˆ − ˆ =− ˆ iωv f u ilgη, (14.78) − ˆ + ˆ =− ˆ iωη iH (ku lv) 0. (14.79) − ˆ + ˆ + ˆ = 11. Gravity Waves with Rotation 637

Solving for u and v between equations (14.77) and (14.78), we obtain ˆ ˆ gη u ˆ (ωk if l), ˆ = ω2 f 2 + − (14.80) gη v ˆ ( if k ωl). ˆ = ω2 f 2 − + − Substituting these in equation (14.79), we obtain

ω2 f 2 gH (k2 l2). (14.81) − = + This is the dispersion relation of gravity waves in the presence of Coriolis forces. (The relation can be most simply derived by setting the determinant of the set of linear homogeneous equations (14.77)–(14.79) to zero.) It can be written as

ω2 f 2 gHK2, (14.82) = + where K √k2 l2 is the magnitude of the horizontal wavenumber. The disper- sion relation= shows+ that the waves can propagate in any horizontal direction and have ω>f. Gravity waves affected by Coriolis forces are called Poincare´ waves, Sverdrup waves, or simply rotational gravity waves. (Sometimes the name “Poincare´ wave” is used to describe those rotational gravity waves that satisfy the boundary conditions in a channel.) In spite of their name, the solution was first worked out by Kelvin (Gill, 1982, p. 197). A plot of equation (14.82) is shown in Figure 14.14. It is seen that the waves are dispersive except for ω f when equation (14.82) gives ω2 gHK2, so that the propagation speed is ω≫/K √gH. The high-frequency limit≃ agrees with our previous discussion of surface= gravity waves unaffected by Coriolis forces.

Figure 14.14 Dispersion relations for Poincare´ and Kelvin waves. 638 Geophysical Fluid Dynamics

Particle Orbit The symmetry of the dispersion relation (14.81) with respect to k and l means that the x- and y-directions are not felt differently by the wavefield. The horizontal isotropy is a result of treating f as constant. (We shall see later that Rossby waves, which depend on the β-effect, are not horizontally isotropic.) We can therefore orient the x-axis along the wavenumber vector and set l 0, so that the wavefield is invariant along the y-axis. To find the particle orbits, it is convenient= to with real quantities. Let the displacement be

η η cos(kx ωt), =ˆ − where η is real. The corresponding velocity components can be found by multiplying equationˆ (14.80) by exp(ikx iωt) and taking the real part of both sides. This gives − ωη u ˆ cos(kx ωt), = kH − (14.83) f η v ˆ sin(kx ωt). = kH − To find the particle paths, take x 0 and consider three values of time corresponding to ωt 0, π/2, and π. The corresponding= values of u and v from equation (14.83) show that= the velocity vector rotates clockwise (in the northern hemisphere) in elliptic paths (Figure 14.15). The ellipticity is expected, since the presence of Coriolis forces means that fu must generate ∂v/∂t according to the equation of motion (14.45). (In equation (14.45), ∂η/∂y 0 due to our orienting the x-axis along the direction of propagation of the wave.)= Particles are therefore constantly deflected to the right by the Coriolis force, resulting in elliptic orbits. The ellipses have an axis ratio of ω/f, and the major axis is oriented in the direction of . The ellipses become narrower as ω/f increases, approaching the rectilinear orbit of gravity waves

Figure 14.15 Particle orbit in a rotational . Velocity components corresponding to ωt 0, = π/2, and π are indicated. 12. 639 unaffected by planetary rotation. However, the sea surface in a rotational gravity wave is no different than that for ordinary gravity waves, namely oscillatory in the direction of propagation and invariant in the perpendicular direction.

Inertial Motion Consider the limit ω f , that is when the particle paths are circular. The dispersion relation (14.82) then shows→ that K 0, implying a horizontal uniformity of the flow field. Equation (14.79) shows that→η must tend to zero in this limit, so that there are no horizontal pressure gradientsˆ in this limit. Because ∂u/∂x ∂v/∂y 0, the continuity equation shows that w 0. The particles therefore move= on horizontal= sheets, each layer decoupled from the= one above and below it. The balance of forces is

∂u fv 0, ∂t − = ∂v fu 0. ∂t + =

The solution of this set is of the form

u q cos ft, = v q sin ft, =− where the speed q √u2 v2 is constant along the path. The radius r of the orbit can be found by adopting= a+ Lagrangian point of view, and noting that the equilibrium of forces is between the Coriolis force fq and the rω2 rf 2, giving r q/f. The limiting case of motion in circular orbits at a frequency= f is called inertial= motion, because in the absence of pressure gradients a particle moves by virtue of its alone. The corresponding period 2π/f is called the inertial period. In the absence of planetary rotation such motion would be along straight lines; in the presence of Coriolis forces the motion is along circular paths, called inertial circles. Near-inertial motion is frequently generated in the surface layer of the ocean by sudden changes of the wind field, essentially because the equations of motion (14.45) have a natural frequency f . Taking a typical current magnitude of q 0.1m/s, the radius of the orbit is r 1 km. ∼ ∼

12. Kelvin Wave In the preceding section we considered a shallow-water gravity wave propagating in a horizontally unbounded ocean. We saw that the crests are horizontal and oriented in adirectionperpendiculartothedirectionofpropagation.Theabsence of a transverse pressure gradient ∂η/∂y resulted in a transverse flow and elliptic orbits. This is clear from the third equation in (14.45), which shows that the presence of fumust result in ∂v/∂t if ∂η/∂y 0. In this section we consider a gravity wave propagating parallel to a wall, whose= presence allows a pressure gradient ∂η/∂y that can decay away from the wall. We shall see that this allows a gravity wave in which fuis geostrophically 640 Geophysical Fluid Dynamics

balanced by g(∂η/∂y), and v 0. Consequently the particle orbits are not elliptic but rectilinear.− = Consider first a gravity wave propagating in a channel. From Figure 7.7 we know that the fluid velocity under a crest is “forward” (i.e., in the direction of propagation), and that under a trough it is backward. Figure 14.16 shows two transverse sections of the wave, one through a crest (left panel) and the other through a trough (right panel). The wave is propagating into the plane of the paper, along the x-direction. Then the fluid velocity under the crest is into the plane of the paper and that under the trough is out of the plane of the paper. The constraints of the side walls require that v 0 at the walls, and we are exploring the possibility of a wave motion in which v is zero= every- where. Then the equation of motion along the y-direction requires that fucan only be geostrophically balanced by a transverse slope of the sea surface across the channel: ∂η fu g . =− ∂y In the northern hemisphere, the surface must slope as indicated in the figure, that is downward to the left under the crest and upward to the left under the trough, so that the pressure force has the current directed to its right. The result is that the amplitude of the wave is larger on the right-hand side of the channel, looking into the direction of propagation, as indicated in Figure 14.16. The current amplitude, like the surface displacement, also decays to the left. If the left wall in Figure 14.16 is moved away to infinity, we get a gravity wave trapped to the coast (Figure 14.17). A coastally trapped long gravity wave, in which the transverse velocity v 0 everywhere, is called a Kelvin wave. It is clear that it can propagate only in a direction= such that the coast is to the right (looking in the direction of propagation) in the northern hemisphere and to the left in the southern hemisphere. The opposite direction of propagation would result in a sea surface displacement increasing exponentially away from the coast, which is not possible. An examination of the transverse momentum equation ∂v ∂η fu g , ∂t + =− ∂y

Figure 14.16 Free surface distribution in a gravity wave propagating through a channel into the plane of the paper. 12. Kelvin Wave 641

Figure 14.17 Coastal Kelvin wave propagating along the x-axis. Sea surface across a section through a crest is indicated by the continuous line, and that along a trough is indicated by the dashed line.

reveals fundamental differences between Poincare´ waves and Kelvin waves. For a Poincare´ wave the crests are horizontal, and the absence of a transverse pressure gradient requires a ∂v/∂t to balance the Coriolis force, resulting in elliptic orbits. In a Kelvin wave a transverse velocity is prevented by a geostrophic balance of fuand g(∂η/∂y). − From the shallow-water set (14.45), the equations of motion for a Kelvin wave propagating along a coast aligned with the x-axis (Figure 14.17) are

∂η ∂u H 0, ∂t + ∂x = ∂u ∂η g , (14.84) ∂t =− ∂x ∂η fu g . =− ∂y

Assume a solution of the form

i(kx ωt) u, η u(y), η(y) e − . [ ]=[ˆ ˆ ] Then equation (14.84) gives iωη iHku 0, − ˆ + ˆ = iωu igkη, (14.85) − ˆ =− ˆ dη f u g ˆ . ˆ =− dy 642 Geophysical Fluid Dynamics

The dispersion relation can be found solely from the first two of these equations; the third equation then determines the transverse structure. Eliminating u between the first two, we obtain ˆ

η ω2 gHk2 0. ˆ[ − ]= A nontrivial solution is therefore possible only if ω k√gH, so that the wave propagates with a nondispersive speed =±

c √gH. (14.86) =

The propagation speed of a Kelvin wave is therefore identical to that of nonrotating gravity waves. Its dispersion equation is a straight line and is shown in Figure 14.14. All frequencies are possible. To determine the transverse structure, eliminate u between the first and third of equation (14.85), giving ˆ

dη f ˆ η 0. dy ± c ˆ = The solution that decays away from the coast is

fy/c η η0 e− , ˆ =

where η0 is the amplitude at the coast. Therefore, the sea surface slope and the velocity field for a Kelvin wave have the form

fy/c η η0 e− cos k(x ct), = − g fy/c (14.87) u η0 e− cos k(x ct), = H − . where we have taken the real parts, and have used equation (14.85) in obtaining the u field. Equations (14.87) show that the transverse decay scale of the Kelvin wave is

c 0 , ≡ f

which is called the Rossby radius of deformation. For a of depth H 5 km, 4 1 = and a midlatitude value of f 10− s− , we obtain c √gH 220 m/s and 0 c/f 2200 km. Tides are= frequently in the form of= coastal Kelvin= waves of semidiurnal= = frequency. The tides are forced by the periodic changes in the gravitational attraction of the moon and the sun. These waves propagate along the boundaries of an ocean basin and cause sea level fluctuations at coastal stations. 12. Kelvin Wave 643

Analogous to the surface or “external” Kelvin waves discussed in the preceding, we can have internal Kelvin waves at the interface between two fluids of different densities (Figure 14.18). If the lower layer is very deep, then the speed of propagation is given by (see equation (7.126))

c g H, = ′ - where H is the thickness of the upper layer and g′ g(ρ2 ρ1)/ρ2 is the reduced gravity. For a continuously stratified medium of depth=H and− buoyancy frequency N, internal Kelvin waves can propagate at any of the normal mode

c NH/nπ, n 1, 2,.... = = The decay scale for internal Kelvin waves, 0 c/f, is called the internal Rossby radius of deformation, whose value is much smaller= than that for the external Rossby radius of deformation. For n 1, a typical value in the ocean is 0 NH/πf 50 km; a typical atmospheric= value is much larger, being of order 0 =1000 km. ∼ Internal Kelvin waves in the ocean are frequently forced by wind changes∼ near coastal areas. For example, a southward wind along the west coast of a continent in the northern hemisphere (say, California) generates an Ekman layer at the ocean surface, in which the mass flow is away from the coast (to the right of the applied wind stress). The mass flux in the near-surface layer is compensated by the movement of deeper water toward the coast, which raises the thermocline. An upward movement of the thermocline, as indicated by the dashed line in Figure 14.18, is called . The vertical movement of the thermocline in the wind-forced region then propagates poleward along the coast as an internal Kelvin wave.

Figure 14.18 Internal Kelvin wave at an interface. Dashed line indicates position of the interface when it is at its maximum height. Displacement of the free surface is much smaller than that of the interface and is oppositely directed. 644 Geophysical Fluid Dynamics

13. Conservation in Shallow-Water Theory In this section we shall derive a useful conservation law for the vorticity of a shal- low layer of fluid. From Section 8, the equations of motion for a shallow layer of homogeneous fluid are

∂u ∂u ∂u ∂η u v fv g , (14.88) ∂t + ∂x + ∂y − =− ∂x ∂v ∂v ∂v ∂η u v fu g , (14.89) ∂t + ∂x + ∂y + =− ∂y ∂h ∂ ∂ (uh) (vh) 0, (14.90) ∂t + ∂x + ∂y =

where h(x, y, t) is the depth of flow and η is the height of the sea surface measured from an arbitrary horizontal plane (Figure 14.19). The x-axis is taken eastward and the y-axis is taken northward, with u and v the corresponding velocity components. The Coriolis frequency f f0 βy is regarded as dependent on latitude. The nonlinear terms have been retained,= including+ those in the continuity equation, which has been written in the form (14.44); note that h H η. We saw in Section 8 that the constant density of the layer and the hydrostatic= pressure+ distribution make the horizontal pressure gradient depth-independent, so that only a depth-independent current can be generated. The vertical velocity is linear in z. A vorticity equation can be derived by differentiating equation (14.88) with respect to y, equation (14.89) with respect to x, and subtracting. The pressure is eliminated, and we obtain

∂ ∂v ∂u ∂ ∂v ∂v ∂ ∂u ∂u u v u v ∂t ∂x − ∂y + ∂x ∂x + ∂y − ∂y ∂x + ∂y # $ ! " ! " ∂u ∂v f0 βv 0. (14.91) + ∂x + ∂y + = # $

Figure 14.19 Shallow layer of instantaneous depth h(x, y, t). 13. Potential Vorticity Conservation in Shallow-Water Theory 645

Following the customary β-plane approximation, we have treated f as constant (and replaced it by an average value f0) except when df/dy appears. Wenow introduce ∂v ∂u ζ , ≡ ∂x − ∂y as the vertical component of relative vorticity, that is, the vorticity measured relative to the rotating earth. Then the nonlinear terms in equation (14.91) can easily be rearranged in the form

∂ζ ∂ζ ∂u ∂v u v ζ. ∂x + ∂y + ∂x + ∂y # $ Equation (14.91) then becomes

∂ζ ∂ζ ∂ζ ∂u ∂v u v (ζ f0) βv 0, ∂t + ∂x + ∂y + ∂x + ∂y + + = # $ which can be written as

Dζ ∂u ∂v (ζ f0) βv 0, (14.92) Dt + + ∂x + ∂y + = # $ where D/Dt is the derivative following the horizontal motion of the layer:

D ∂ ∂ ∂ u v . Dt ≡ ∂t + ∂x + ∂y

The horizontal divergence (∂u/∂x ∂v/∂y) in equation (14.92) can be eliminated by using the continuity equation (14.90),+ which can be written as

Dh ∂u ∂v h 0. Dt + ∂x + ∂y = # $ Equation (14.92) then becomes

Dζ ζ f0 Dh + βv. Dt = h Dt − This can be written as

D(ζ f) ζ f0 Dh + + , (14.93) Dt = h Dt where we have used

Df ∂f ∂f ∂f u v vβ. Dt = ∂t + ∂x + ∂y = 646 Geophysical Fluid Dynamics

Because of the absence of vertical shear, the vorticity in a shallow-water model is purely vertical and independent of depth. The relative vorticity measured with respect to the rotating earth is ζ, while f is the planetary vorticity, so that the absolute vorticity is (ζ f ). Equation (14.93) shows that the rate of change of absolute vorticity is proportional+ to the absolute vorticity times the vertical stretching Dh/Dt of the water column. It is apparent that Dζ/Dt can be nonzero even if ζ 0 initially. This is different from a nonrotating flow in which stretching a fluid line changes= its vorticity only if the line has an initial vorticity. (This is why the process was called the vortex stretching; see Chapter 5, Section 7.) The difference arises because vertical lines in a rotating earth contain the planetary vorticity even when ζ 0. Note that the vortex tilting term, discussed in Chapter 5, Section 7, is absent in= the shallow-water theory because the water moves in the form of vertical columns without ever tilting. Equation (14.93) can be written in the compact form

D ζ f + 0, (14.94) Dt h = # $

where f f0 βy, and we have assumed βy f0. The ratio (ζ f )/h is called the potential= vorticity+ in shallow-water theory. Equation≪ (14.94) shows+ that the potential vorticity is conserved along the motion, an important principle in geophysical fluid dynamics. In the ocean, outside regions of strong current vorticity such as coastal boundaries, the magnitude of ζ is much smaller than that of f . In such a case (ζ f ) has the sign of f . The principle of conservation of potential vorticity means that+ an increase in h must make (ζ f ) more positive in the northern hemisphere and more negative in the southern hemisphere.+ As an example of application of the potential vorticity equation, consider an eastward flow over a step (at x 0) running north–south, across which the layer = thickness changes discontinuously from h0 to h1 (Figure 14.20). The flow upstream of the step has a uniform speed U, so that the oncoming stream has no relative vorticity. To conserve the ratio (ζ f )/h,theflowmustsuddenlyacquirenegative(clockwise) relative vorticity due to+ the sudden decrease in layer thickness. The relative vorticity of a fluid element just after passing the step can be found from f ζ f + , h0 = h1

giving ζ f (h1 h0)/h0 < 0, where f is evaluated at the upstream latitude of the streamline.= Because− of the clockwise vorticity, the fluid starts to move south at x 0. The southward movement decreases f , so that ζ must correspondingly increase= so as to keep (f ζ ) constant. This means that the clockwise curvature of the stream reduces, and eventually+ becomes a counterclockwise curvature. In this manner an eastward flow over a step generates stationary undulatory flow on the downstream side. In Section 15 we shall see that the stationary oscillation is due to a Rossby wave generated at the step whose westward is canceled by the eastward current. We shall see that the wavelength is 2π√U/β. 14. Internal Waves 647

Figure 14.20 Eastward flow over a step, resulting in stationary oscillations of wavelength 2π√U/β.

Suppose we try the same argument for a westward flow over a step. Then a particle should suddenly acquire clockwise vorticity as the depth of flow decreases at x 0, which would require the particle to move north. It would then come into a region= of larger f , which would require ζ to decrease further. Clearly, an exponential behavior is predicted, suggesting that the argument is not correct. Unlike an eastward flow, a westward current feels the upstream influence of the step so that it acquires a counterclockwise curvature before it encounters the step (Figure 14.21). The positive vorticity is balanced by a reduction in f , which is consistent with conservation of potential vorticity. At the location of the step the vorticity decreases suddenly. Finally, far downstream of the step a fluid particle is again moving westward at its original latitude. The westward flow over a topography is not oscillatory.

14. Internal Waves In Chapter 7, Section 19 we studied internal gravity waves unaffected by Cori- olis forces. We saw that they are not isotropic; in fact the direction of propa- gation with respect to the vertical determines their frequency. We also saw that their frequency satisfies the inequality ω " N, where N is the buoyancy frequency. Their phase-velocity vector c and the group-velocity vector cg are perpendicular and have oppositely directed vertical components (Figure 7.32 and Figure 7.34). That is, phases propagate upward if the groups propagate downward, and vice versa. In this section we shall study the effect of Coriolis forces on internal waves, assuming that f is independent of latitude. Internal waves are ubiquitous in the atmosphere and the ocean. In the lower atmo- sphere turbulent motions dominate, so that activity represents a minor component of the motion. In contrast, the stratosphere contains very little convective motion because of its stable density distribution, and consequently a great deal of internal wave activity. They generally propagate upward from the lower atmosphere, where they are generated. In the ocean they may be as common as the waves on 648 Geophysical Fluid Dynamics

Figure 14.21 Westward flow over a step. Unlike the eastward flow, the westward flow is not oscillatory and feels the upstream influence of the step.

the surface, and measurements show that they can cause the isotherms to go up and down by as much as 50–100 m. Sometimes the internal waves break and generate small-scale turbulence, similar to the “foam” generated by breaking waves. We shall now examine the nature of the fluid motion in internal waves. The equations of motion are

∂u ∂v ∂w 0, ∂x + ∂y + ∂z = ∂u 1 ∂p fv , ∂t − =−ρ0 ∂x ∂v 1 ∂p fu , (14.95) ∂t + =−ρ0 ∂y ∂w 1 ∂p ρg , ∂t =−ρ0 ∂z − ρ0 ∂ρ ρ N 2 0 w 0. ∂t − g =

We have not made the hydrostatic assumption because we are not assuming that the horizontal wavelength is long compared to the vertical wavelength. The advective 2 term in the density equation is written in a linearized form w(dρ/dz) ρ0N w/g. Thus the rate of change of density at a point is assumed to be¯ due only=− to the ver- tical advection of the background density distribution ρ(z). Because internal wave activity is more intense in the thermocline where N varies¯ appreciably (Figure 14.2), we shall be somewhat more general than in Chapter 7 and not assume that N is depth-independent. An equation for w can be formed from the set (14.95) by eliminating all other variables. The algebraic steps of such a procedure are shown in Chapter 7, Section 18 without the Coriolis forces. This gives 14. Internal Waves 649

∂2 ∂2w 2w N 2 2 w f 2 0, (14.96) ∂t2 ∇ + ∇H + ∂z2 = where 2 2 2 2 ∂ ∂ ∂ ∇ ≡ ∂x2 + ∂y2 + ∂z2 and ∂2 ∂2 2 . ∇H ≡ ∂x2 + ∂y2 Because the coefficients of equation (14.96) are independent of the horizontal direc- tions, equation (14.96) can have solutions that are trigonometric in x and y.We therefore assume a solution of the form

i(kx ly ωt) u, v, w u(z), v(z), w(z) e + − . (14.97) [ ]=[ˆ ˆ ˆ ] Substitution into equation (14.96) gives

d2 d2w ( iω)2 (ik)2 (il)2 w N 2 (ik)2 (il)2 w f 2 ˆ 0, − + + dz2 ˆ + [ + ]ˆ + dz2 = ! " from which we obtain d2w (N 2 ω2)(k2 l2) ˆ − + w 0. (14.98) dz2 + ω2 f 2 ˆ = − Defining

(k2 l2) N 2(z) ω2 m2(z) + [ − ], (14.99) ≡ ω2 f 2 − Equation (14.98) becomes

d2w ˆ m2w 0. (14.100) dz2 + ˆ =

For m2 < 0, the solutions of equation (14.100) are exponential in z signifying that the resulting motion is surface-trapped. It represents a propagating hor- izontally. For a positive m2, on the other hand, solutions are trigonometric in z, giving internal waves propagating vertically as well as horizontally. From equation (14.99), therefore, internal waves are possible only in the frequency range:

f<ωf, as is true for much of the atmosphere and the ocean. 650 Geophysical Fluid Dynamics

WKB Solution To proceed further, we assume that N(z) is a slowly varying function in that its fractional change over a vertical wavelength is much less than unity. We are therefore considering only those internal waves whose vertical wavelength is short compared to the scale of variation of N. If H is a characteristic vertical distance over which N varies appreciably, then we are assuming that

Hm 1. ≫ For such slowly varying N(z), we expect that m(z) given by equation (14.99) is also a slowly varying function, that is, m(z) changes by a small fraction in a distance 1/m. Under this assumption the waves locally behave like plane waves, as if m is constant. This is the so-called WKB approximation (after Wentzel–Kramers–Brillouin), which applies when the properties of the medium (in this case N) are slowly varying. To derive the approximate WKB solution of equation (14.100), we look for a solution in the form

w A(z)eiφ(z), ˆ = where the phase φ and the (slowly varying) amplitude A are real. (No generality is lost by assuming A to be real. Suppose it is complex and of the form A A exp(iα), ¯ where A and α are real. Then w A exp i(φ α) , a form in which (=φ α) is the ¯ ¯ phase.) Substitution into equationˆ = (14.100)[ gives+ ] +

d2A dφ 2 dA dφ d2φ A m2 i2 iA 0. dz2 + − dz + dz dz + dz2 = ! # $ " Equating the real and imaginary parts, we obtain

d2A dφ 2 A m2 0, (14.101) dz2 + − dz = ! # $ " dA dφ d2φ 2 A 0. (14.102) dz dz + dz2 =

In equation (14.101) the term d2A/dz2 is negligible because its ratio with the second term is

d2A/dz2 1 1. Am2 ∼ H 2m2 ≪

Equation (14.101) then becomes approximately dφ m, (14.103) dz =± 14. Internal Waves 651 whose solution is z φ mdz, =± ) the lower limit of the integral being arbitrary. The amplitude is determined by writing equation (14.102) in the form

dA (d2φ/dz2)dz (dm/dz) dz 1 dm , A =− 2(dφ/dz) =− 2m =−2 m

1 where equation (14.103) has been used. Integrating, we obtain ln A 2 ln m const., that is, =− +

A A 0 , = √m where A0 is a constant. The WKB solution of equation (14.100) is therefore

A0 i z mdz w e± . (14.104) ˆ = √m * Because of neglect of the β-effect, the waves must behave similarly in x and y, as indicated by the symmetry of the dispersion relation (14.99) in k and l. Therefore, we lose no generality by orienting the x-axis in the direction of propagation, and taking

k>0 l 0 ω>0. = To find u and v in terms of w, use the continuity equation ∂u/∂x ∂w/∂z 0, noting that the y-derivatives are zero because of our setting l 0. Substituting+ = the wave solution (14.97) into the continuity equation gives =

dw iku ˆ 0. (14.105) ˆ + dz = The z-derivative of w in equation (14.104) can be obtained by treating the denominator √m as approximatelyˆ constant because the variation of w is dominated by the wiggly behavior of the local plane wave solution. This gives ˆ

dw A0 i z mdz i z mdz ˆ ( im)e± iA0√me± , dz = √m ± * =± * so that equation (14.105) becomes

A0√m i z mdz u e± . (14.106) ˆ =∓ k * 652 Geophysical Fluid Dynamics

An expression for v can now be obtained from the horizontal equations of motion in equation (14.95). Crossˆ differentiating, we obtain the vorticity equation

∂ ∂u ∂v ∂u ∂v f . ∂t ∂y − ∂x = ∂x + ∂y # $ # $ Using the wave solution equation (14.97), this gives

u iω ˆ . v = f ˆ Equation (14.106) then gives

if A0√m i z mdz v e± . (14.107) ˆ =±ω k * Taking real parts of equations (14.104), (14.106), and (14.107), we obtain the velocity field

A √m z u 0 cos kx mdz ωt , =∓ k ± − # $ ) z A0f √m v sin kx mdz ωt , (14.108) =∓ ωk ± − # ) $ A z w 0 cos kx mdz ωt , = √m ± − # ) $ where the dispersion relation is

k2(N 2 ω2) m2 − . (14.109) = ω2 f 2 −

The meaning of m(z) is clear from equation (14.108). If we call the argument of the trigonometric terms the “phase,” then it is apparent that ∂(phase)/∂z m(z), so that m(z) is the local vertical wavenumber. Because we are treating k, m, ω=>0, it is also apparent that the upper signs represent waves with upward phase propagation, and the lower signs represent downward phase propagation.

Particle Orbit To find the shape of the hodograph in the horizontal plane, consider the point x z 0. Then equation (14.108) gives = = u cos ωt, =∓ f (14.110) v sin ωt, =±ω 14. Internal Waves 653

Figure 14.22 Particle orbit in an internal wave. The upper panel (a) shows projection on a horizontal plane; points corresponding to ωt 0, π/2, and π are indicated. The lower panel (b) shows a three-dimensional = view. Sense of rotation shown is valid for the northern hemisphere. where the amplitude of u has been arbitrarily set to one. Taking the upper signs in equation (14.110), the values of u and v are indicated in Figure 14.22a for three values of time corresponding to ωt 0, π/2, and π. It is clear that the horizontal hodographs are clockwise ellipses, with= the major axis in the direction of propagation x, and the axis ratio is f/ω. The same conclusion applies for the lower signs in equation (14.110). The particle orbits in the horizontal plane are therefore identical to those of Poincare´ waves (Figure 14.15). However, the plane of the motion is no longer horizontal. From the velocity components equation (14.108), we note that

u m tan θ, (14.111) w =∓k =∓

1 where θ tan− (m/k) is the angle made by the wavenumber vector K with the horizontal= (Figure 14.23). For upward phase propagation, equation (14.111) gives u/w tan θ, so that w is negative if u is positive, as indicated in Figure 14.23. A three-dimensional=− sketch of the particle orbit is shown in Figure 14.22b. It is easy to show (Exercise 6) that the phase velocity vector c is in the direction of K, that c and cg are perpendicular, and that the fluid motion u is parallel to cg; these facts are demonstrated in Chapter 7 for internal waves unaffected by Coriolis forces. 654 Geophysical Fluid Dynamics

Figure 14.23 Vertical section of an internal wave. The three parallel lines are constant phase lines, with the arrows indicating fluid motion along the lines.

The velocity vector at any location rotates clockwise with time. Because of the verticalpropagationofphase,thetipsofthe instantaneous vectors also turn with depth. Consider the turning of the velocity vectors with depth when the phase velocity is upward, so that the deeper currents have a phase lead over the shallower currents (Figure 14.24). Because the currents at all depths rotate clockwise in time (whether the vertical component of c is upward or downward), it follows that the tips of the instantaneous velocity vectors should fall on a helical spiral that turns clockwise with depth. Only such a turning in depth, coupled with a clockwise rotation of the velocity vectors with time, can result in a phase lead of the deeper currents. In the opposite case of a downward phase propagation, the helix turns counterclockwise with depth. The direction of turning of the velocity vectors can also be found from equation (14.108), by considering x t 0 and finding u and v at various values of z. = = Discussion of the Dispersion Relation The dispersion relation (14.109) can be written as

k2 ω2 f 2 (N 2 ω2). (14.112) − = m2 − Introducing tan θ m/k, equation (14.112) becomes = ω2 f 2 sin2θ N 2 cos2θ, = + which shows that ω is a function of the angle made by the wavenumber with the horizontal and is not a function of the magnitude of K. For f 0 the forementioned expression reduces to ω N cos θ, derived in Chapter 7, Section= 19 without Coriolis forces. = 14. Internal Waves 655

Figure 14.24 Helical spiral traced out by the tips of instantaneous velocity vectors in an internal wave with upward phase speed. Heavy arrows show the velocity vectors at two depths, and light arrows indicate that they are rotating clockwise with time. Note that the instantaneous vectors turn clockwise with depth.

Figure 14.25 Dispersion relation for internal waves. The different regimes are indicated on the left-hand side of the figure.

A plot of the dispersion relation (14.112) is presented in Figure 14.25, showing ω as a function of k for various values of m. All curves pass through the point ω f , which represents inertial oscillations. Typically, N f in most of the atmosphere= and the ocean. Because of the wide separation of the≫ upper and lower limits of the internal wave range f " ω " N, various limiting cases are possible, as indicated in Figure 14.25. They are 2 (1) High-frequency regime (ω N, but ω " N): In this range f is negligible in comparison with ω2 in the∼ denominator of the dispersion relation (14.109), which reduces to 656 Geophysical Fluid Dynamics

k2(N 2 ω2) N 2k2 m2 − , that is,ω2 . ≃ ω2 ≃ m2 k2 + Using tan θ m/k, this gives ω N cos θ. Thus, the high-frequency inter- nal waves are= the same as the= nonrotating internal waves discussed in Chapter 7. 2 (2) Low-frequency regime (ω f, but ω ! f ): In this range ω can be neglected in comparison to N 2 in the∼ dispersion relation (14.109), which becomes

k2N 2 k2N 2 m2 , that is,ω2 f 2 . ≃ ω2 f 2 ≃ + m2 − The low-frequency limit is obtained by making the hydrostatic assumption, that is, neglecting ∂w/∂t in the vertical equation of motion. (3) Midfrequency regime (f ω N): In this range the dispersion relation (14.109) simplifies to ≪ ≪

k2N 2 m2 , ≃ ω2 so that both the hydrostatic and the nonrotating assumptions are applicable.

Lee Wave Internal waves are frequently found in the “lee” (that is, the downstream side) of . In stably stratified conditions, the flow of air over a mountain causes a vertical displacement of fluid particles, which sets up internal waves as it moves downstream of the mountain. If the amplitude is large and the air is moist, the upward motion causes condensation and cloud formation. Due to the effect of a mean flow, the lee waves are stationary with respect to the ground. This is shown in Figure 14.26, where the westward phase speed is canceled by the eastward mean flow. We shall determine what wave parameters make this cancellation possible. The frequency of lee waves is much larger than f , so that rotational effects are negligible. The dispersion relation is therefore

N 2k2 ω2 . (14.113) = m2 k2 + However, we now have to introduce the effects of the mean flow. The dispersion relation (14.113) is still valid if ω is interpreted as the intrinsic frequency, that is, the frequency measured in a moving with the mean flow. In a medium moving with a velocity U, the observed frequency of waves at a fixed point is Doppler shifted to

ω0 ω K U, = + • 15. Rossby Wave 657

Figure 14.26 Streamlines in a lee wave. The thin line drawn through crests shows that the phase propa- gates downward and westward. where ω is the intrinsic frequency; this is discussed further in Chapter 7, Section 3. For a stationary wave ω0 0, which requires that the intrinsic frequency is = ω K • U kU. (Here K • U is positive because K is westward and U is eastward.)=− The= dispersion relation− (14.113) then gives N U . = √k2 m2 + If the flow speed U is given, and the mountain introduces a typical horizontal wavenumber k, then the preceding equation determines the vertical wavenumber m that generates stationary waves. Waves that do not satisfy this condition would radiate away. The source of lee waves is at the surface. The energy therefore must prop- agate upward, and consequently the phases propagate downward. The intrinsic phase speed is therefore westward and downward in Figure 14.26. With this information, we can determine which way the constant phase lines should tilt in a stationary lee wave. Note that the wave pattern in Figure 14.26 would propagate to the left in the absence of a mean velocity, and only with the constant phase lines tilting backwards with height would the flow at larger height lead the flow at a lower height. Further discussion of internal waves can be found in Phillips (1977) and Munk (1981); lee waves are discussed in Holton (1979).

15. Rossby Wave To this point we have discussed wave motions that are possible with a constant Coriolis frequency f and found that these waves have frequencies larger than f . We shall now consider wave motions that owe their existence to the variation of f with latitude. With such a variable f , the equations of motion allow a very important type of wave motion called the Rossby wave. Their spatial scales are so large in the atmosphere that they usually have only a few wavelengths around the entire globe (Figure 14.27). This 658 Geophysical Fluid Dynamics

Figure 14.27 Observed height (in decameters) of the 50 kPa pressure surface in the northern hemi- sphere. The center of the picture represents the north pole. The undulations are due to Rossby waves (dm km/100). J. T. Houghton, The of the Atmosphere, 1986 and reprinted with the permission = of Cambridge University Press.

is why Rossby waves are also called planetary waves. In the ocean, however, their wavelengths are only about 100 km. Rossby-wave frequencies obey the inequality ω f . Because of this slowness the time derivative terms are an order of magnitude smaller≪ than the Coriolis forces and the pressure gradients in the horizontal equations of motion. Such nearly geostrophic flows are called quasi-geostrophic motions.

Quasi-Geostrophic Vorticity Equation We shall first derive the governing equation for quasi-geostrophic motions. For sim- plicity, we shall make the customary β-plane approximation valid for βy f0, keep- ing in mind that the approximation is not a good one for atmospheric Rossby≪ waves, which have planetary scales. Although Rossby waves are frequently superposed on a mean flow, we shall derive the equations without a mean flow, and superpose a uniform mean flow at the end, assuming that the perturbations are small and that a linear superposition is valid. The first step is to simplify the vorticity equation for quasi-geostrophic motions, assuming that the velocity is geostrophic to the lowest order.Thesmalldeparturesfromgeostrophy,however,areimportantbecausethey determine the evolution of the flow with time. 15. Rossby Wave 659

We start with the shallow-water potential vorticity equation

D ζ f + 0, Dt h = # $ which can be written as D Dh h (ζ f) (ζ f) 0. Dt + − + Dt = We now expand the material derivative and substitute h H η, where H is the uniform undisturbed depth of the layer, and η is the surface= displacement.+ This gives

∂ζ ∂ζ ∂ζ ∂η ∂η ∂η (H η) u v βv (ζ f0) u v 0. + ∂t + ∂x + ∂y + − + ∂t + ∂x + ∂y = # $ # $ (14.114)

Here, we have used Df/Dt v(df/dy) βv. We have also replaced f by f0 in the second term because the= β-plane approximation= neglects the variation of f except when it involves df/dy. For small perturbations we can neglect the quadratic nonlinear terms in equation (14.114), obtaining ∂ζ ∂η H Hβv f0 0. (14.115) ∂t + − ∂t = This is the linearized form of the potential vorticity equation. Its quasi-geostrophic ver- sion is obtained if we substitute the approximate geostrophic expressions for velocity: g ∂η u , ≃−f ∂y 0 (14.116) g ∂η v . ≃ f0 ∂x From this the vorticity is found as

g ∂2η ∂2η ζ , = f ∂x2 + ∂y2 0 # $ so that the vorticity equation (14.115) becomes

gH ∂ ∂2η ∂2η gHβ ∂η ∂η f0 0. f ∂t ∂x2 + ∂y2 + f ∂x − ∂t = 0 # $ 0 Denoting c √gH, this becomes = 2 2 2 ∂ ∂ η ∂ η f0 ∂η 2 2 2 η β 0. (14.117) ∂t /∂x + ∂y − c 0 + ∂x = 660 Geophysical Fluid Dynamics

This is the quasi-geostrophic form of the linearized vorticity equation, which governs the flow of large-scale motions. The ratio c/f0 is recognized as the Rossby radius. Note that we have not set ∂η/∂t 0, in equation (14.115) during the derivation of equation (14.117), although a strict= validity of the geostrophic relations (14.116) would require that the horizontal divergence, and hence ∂η/∂t, be zero. This is because the departure from strict geostrophy determines the evolution of the flow described by equation (14.117). We can therefore use the geostrophic relations for velocity everywhere except in the horizontal divergence term in the vorticity equation.

Dispersion Relation Assume solutions of the form

i(kx ly ωt) η ηe + − . =ˆ We shall regard ω as positive; the signs of k and l then determine the direction of phase propagation. A substitution into the vorticity equation (14.117) gives

βk ω . (14.118) =−k2 l2 f 2/c2 + + 0

This is the dispersion relation for Rossby waves. The asymmetry of the dispersion relation with respect to k and l signifies that the wave motion is not isotropic in the horizontal, which is expected because of the β-effect. Although we have derived it for a single homogeneous layer, it is equally applicable to stratified flows if c is replaced by the corresponding internal value, which is c g′H for the reduced gravity model (see Chapter 7, Section 17) and c NH/n=π for the nth mode of a = - 2 2 continuously stratified model. For the barotropic mode c is very large, and f0 /c is usually negligible in the denominator of equation (14.118). The dispersion relation ω(k, l) in equation (14.118) can be displayed as a surface, taking k and l along the horizontal axes and ω along the vertical axis. The section of this surface along l 0 is indicated in the upper panel of Figure 14.28, and sections of the surface for three= values of ω are indicated in the bottom panel. The contours of constant ω are circles because the dispersion relation (14.118) can be written as

β 2 β 2 f 2 k l2 0 . + 2ω + = 2ω − c2 # $ # $ The definition of ∂ω ∂ω cg i j , = ∂k + ∂l shows that the group velocity vector is the gradient of ωin the wavenumber space. The direction of cg is therefore perpendicular to the ω contours, as indicated in the lower 15. Rossby Wave 661

Figure 14.28 Dispersion relation ω(k, l) for a Rossby wave. The upper panel shows ω vs k for l 0. = Regions of positive and negative group velocity cgx are indicated. The lower panel shows a plan view of the surface ω(k, l), showing contours of constant ω on a kl-plane. The values of ωf0/βc for the three circles are 0.2, 0.3, and 0.4. Arrows perpendicular to ω contours indicate directions of group velocity vector cg. A. E. Gill, Atmosphere–, 1982 and reprinted with the permission of Academic Press and Mrs. Helen Saunders-Gill. panel of Figure 14.28. For l 0, the maximum frequency and zero group speed = are attained at kc/f0 1, corresponding to ωmaxf0/βc 0.5. The maximum frequency is much smaller=− than the Coriolis frequency. For= example, in the ocean 2 the ratio ωmax/f 0 0.5βc/f0 is of order 0.1 for the barotropic mode, and of order = 4 1 0.001 for a baroclinic mode, taking a typical midlatitude value of f0 10 s , ∼ − − 662 Geophysical Fluid Dynamics

a barotropic gravity wave speed of c 200 m/s, and a baroclinic gravity wave speed of c 2m/s. The shortest period of midlatitude∼ baroclinic Rossby waves in the ocean can therefore∼ be more than a year. The eastward phase speed is ω β cx . (14.119) = k =−k2 l2 f 2/c2 + + 0 The negative sign shows that the phase propagation is always westward. The phase speed reaches a maximum when k2 l2 0, corresponding to very large wavelengths represented by the region near the origin+ → of Figure 14.28. In this region the waves are nearly nondispersive and have an eastward phase speed

βc2 cx 2 . ≃−f0

11 1 1 With β 2 10− m− s− , a typical baroclinic value of c 2m/s, and a mid- = × 4 1 2 ∼ latitude value of f0 10− s− , this gives cx 10− m/s. At these slow speeds the Rossby waves would∼ take years to cross the width∼ of the ocean at midlatitudes. The Rossby waves in the ocean are therefore more important at lower , where they propagate faster. (The dispersion relation (14.118), however, is not valid within a latitude band of 3◦ from the equator, for then the assumption of a near geostrophic balance breaks down. A different analysis is needed in the tropics. A discussion of the wave dynamics of the tropics is given in Gill (1982) and in the review paper by McCreary (1985).) In the atmosphere c is much larger, and consequently the Rossby waves propagate faster. A typical large atmospheric disturbance can propagate as a Rossby wave at a speed of several meters per second. Frequently, the Rossby waves are superposed on a strong eastward mean current, such as the atmospheric . If U is the speed of this eastward current, then the observed eastward phase speed is β cx U . (14.120) = − k2 l2 f 2/c2 + + 0 Stationary Rossby waves can therefore form when the eastward current cancels the westward phase speed, giving cx 0. This is how stationary waves are formed down- stream of the topographic step in Figure= 14.20. A simple expression for the wavelength results if we assume l 0 and the flow is barotropic, so that f 2/c2 is negligible in = 0 equation (14.120). This gives U β/k2 for stationary solutions, so that the wave- length is 2π√U/β. = Finally, note that we have been rather cavalier in deriving the quasi-geostrophic vorticity equation in this section, in the sense that we have substituted the approximate geostrophic expressions for velocity without a formal ordering of the scales. Gill (1982) has given a more precise derivation, expanding in terms of a small parameter. Another way to justify the dispersion relation (14.118) is to obtain it from the general dispersion relation (14.76) derived in Section 10: 16. Barotropic Instability 663

ω3 c2ω(k2 l2) f 2ω c2βk 0. (14.121) − + − 0 − = For ω f , the first term is negligible compared to the third, reducing equation≪ (14.121) to equation (14.118).

16. Barotropic Instability In Chapter 12, Section 9 we discussed the inviscid stability of a shear flow U(y) in a nonrotating system, and demonstrated that a necessary condition for its instability is that d2U/dy2 must change sign somewhere in the flow. This was called Rayleigh’s point of inflection criterion. In terms of vorticity ζ dU/dy, the criterion states ¯ that dζ/dy must change sign somewhere in the flow.= We− shall now show that, on a ¯ rotating earth, the criterion requires that d(ζ f )/dy must change sign somewhere ¯ within the flow. + Consider a horizontal current U(y) in a medium of uniform density. In the absence of horizontal density gradients only the barotropic mode is allowed, and U(y) does not vary with depth. The vorticity equation is

∂ u (ζ f) 0. (14.122) ∂t + • ∇ + = # $ This is identical to the potential vorticity equation D/Dt (ζ f )/h 0, with the added simplification that the layer depth is constant because[ w+ 0. Let]= the total flow be decomposed into background flow plus a disturbance: =

u U(y) u′, = + v v′. = The total vorticity is then

dU ∂v′ ∂u′ dU 2 ζ ζ ζ ′ ψ, = ¯ + =−dy + ∂x − ∂y =−dy +∇ # $ where we have defined the perturbation streamfunction ∂ψ ∂ψ u′ ,v′ . =−∂y = ∂x Substituting into equation (14.122) and linearizing, we obtain the perturbation vor- ticity equation

∂ ∂ d2U ∂ψ ( 2ψ) U ( 2ψ) β 0. (14.123) ∂t ∇ + ∂x ∇ + − dy2 ∂x = # $ 664 Geophysical Fluid Dynamics

Because the coefficients of equation (14.123) are independent of x and t, there can be solutions of the form

ik(x ct) ψ ψ(y) e − . = ˆ

The phase speed c is complex and solutions are unstable if its imaginary part ci > 0. The perturbation vorticity equation (14.123) then becomes

d2 d2U (U c) k2 ψ β ψ 0. − dy2 − ˆ + − dy2 ˆ = ! " ! " Comparing this with equation (12.76) derived without Coriolis forces, it is seen that the effect of planetary rotation is the replacement of d2U/dy2 by (β d2U/dy2). The analysis of the section therefore carries over to the− present case, resulting− in the following criterion: A necessary condition for the inviscid instability of a barotropic current U(y) is that the gradient of the absolute vorticity

d d2U (ζ f) β , (14.124) dy ¯ + = − dy2 must change sign somewhere in the flow. This result was first derived by Kuo (1949). Barotropic instability quite possibly plays an important role in the instability of currents in the atmosphere and in the ocean. The instability has no preference for any latitude, because the criterion involves β and not f . However, the mechanism presum- ably dominates in the tropics because midlatitude disturbances prefer the baroclinic instability mechanism discussed in the following section. An unstable distribution of westward tropical wind is shown in Figure 14.29.

Figure 14.29 Profiles of velocity and vorticity of a westward tropical wind. The velocity distribution is barotropically unstable as d(ζ f )/dy changes sign within the flow. J. T. Houghton, The Physics of the ¯ + Atmosphere, 1986 and reprinted with the permission of Cambridge University Press. 17. Baroclinic Instability 665

17. Baroclinic Instability The weather maps at midlatitudes invariably show the presence of wavelike horizontal excursions of temperature and pressure contours, superposed on eastward mean flows such as the jet stream. Similar undulations are also found in the ocean on eastward currents such as the Gulf Stream in the north Atlantic. A typical wavelength of these disturbances is observed to be of the order of the internal Rossby radius, that is, about 4000 km in the atmosphere and 100 km in the ocean. They seem to be propagating as Rossby waves, but their erratic and unexpected appearance suggests that they are not forced by any external agency, but are due to an inherent instability of midlatitude eastward flows. In other words, the eastward flows have a spontaneous tendency to develop wavelike disturbances. In this section we shall investigate the instability mechanism that is responsible for the spontaneous relaxation of eastward jets into a meandering state. The poleward decrease of the solar irradiation results in a poleward decrease of the temperature and a consequent increase of the density. An idealized distribution of the atmospheric density in the northern hemisphere is shown in Figure 14.30. The den- sity increases northward due to the lower near the poles and decreases upward because of static stability. According to the thermal wind relation (14.15), an eastward flow (such as the jet stream in the atmosphere or the Gulf Stream in the Atlantic) in equilibrium with such a density structure must have a velocity that increases with height. A system with inclined density surfaces, such as the one in Figure 14.30, has more than a system with horizontal density sur- faces, just as a system with an inclined free surface has more potential energy than a system with a horizontal free surface. It is therefore potentially unstable because it can release the stored potential energy by means of an instability that would cause the density surfaces to flatten out. In the process, vertical shear of the mean flow U(z) would decrease, and perturbations would gain . Instability of baroclinic jets that release potential energy by flattening out the density surfaces is called the baroclinic instability. Our analysis would show that the preferred scale of the unstable waves is indeed of the order of the Rossby radius, as observed for the midlatitude weather disturbances. The theory of baroclinic instability

Figure 14.30 Lines of constant density in the northern hemispheric atmosphere. The lines are nearly horizontal and the slopes are greatly exaggerated in the figure. The velocity U(z) is into the plane of paper. 666 Geophysical Fluid Dynamics

was developed in the 1940s by Bjerknes et al. and is considered one of the major triumphs of geophysical fluid mechanics. Our presentation is essentially based on the review article by Pedlosky (1971). Consider a basic state in which the density is stably stratified in the vertical with a uniform buoyancy frequency N, and increases northward at a constant rate ∂ρ/∂y. According to the thermal wind relation, the constancy of ∂ρ/∂y requires that the¯ vertical shear of the basic eastward flow U(z) also be constant.¯ The β-effect is neglected as it is not an essential requirement of the instability. (The β-effect does modify the instability, however.) This is borne out by the spontaneous appearance of undulations in laboratory experiments in a rotating annulus, in which the inner wall is maintained at a higher temperature than the outer wall. The β-effect is absent in such an experiment.

Perturbation Vorticity Equation The equations for total flow are ∂u ∂u ∂u 1 ∂p u v fv , ∂t + ∂x + ∂y − =−ρ0 ∂x ∂v ∂v ∂v 1 ∂p u v fu , ∂t + ∂x + ∂y + =−ρ0 ∂y ∂p 0 ρg, (14.125) =−∂z − ∂u ∂v ∂w 0, ∂x + ∂y + ∂z = ∂ρ ∂ρ ∂ρ ∂ρ u v w 0, ∂t + ∂x + ∂y + ∂z =

where ρ0 is a constant reference density. We assume that the total flow is composed of a basic eastward jet U(z) in geostrophic equilibrium with the basic density structure ρ(y, z) shown in Figure 14.30, plus perturbations. That is, ¯ u U(z) u′(x, y, z), = + v v′(x, y, z), = w w′(x, y, z), (14.126) = ρ ρ(y, z) ρ′(x, y, z), =¯ + p p(y, z) p′(x, y, z). =¯ + The basic flow is in geostrophic and hydrostatic balance: 1 ∂p fU ¯ , =−ρ0 ∂y (14.127) ∂p 0 ¯ ρg. =−∂z −¯ 17. Baroclinic Instability 667

Eliminating the pressure, we obtain the thermal wind relation dU g ∂ρ ¯ , (14.128) dz = fρ0 ∂y which states that the eastward flow must increase with height because ∂ρ/∂y > 0. For simplicity, we assume that ∂ρ/∂y is constant, and that U 0 at the surface¯ z 0. Thus the background flow is ¯ = = U z U 0 , = H where U0 is the velocity at the top of the layer at z H . We first form a vorticity equation by cross differentiating= the horizontal equations of motion in equation (14.125), obtaining ∂ζ ∂ζ ∂ζ ∂w u v (ζ f) 0. (14.129) ∂t + ∂x + ∂y − + ∂z = This is identical to equation (14.92), except for the exclusion of the β-effect here; the algebraic steps are therefore not repeated. Substituting the decomposition (14.126), and noting that ζ ζ ′ because the basic flow U U0z/H has no vertical component of vorticity, (14.129)= becomes =

∂ζ ∂ζ ∂w ′ U ′ f ′ 0, (14.130) ∂t + ∂x − ∂z = where the nonlinear terms have been neglected. This is the perturbation vorticity equation, which we shall now write in terms of p′. Assume that the perturbations are large-scale and slow, so that the velocity is nearly geostrophic:

1 ∂p′ 1 ∂p′ u′ ,v′ , (14.131) ≃−ρ0f ∂y ≃ ρ0f ∂x from which the perturbation vorticity is found as

1 2 ζ ′ Hp′. (14.132) = ρ0f ∇

We now express w′ in equation (14.130) in terms of p′. The density equation gives ∂ ∂ ∂ ∂ (ρ ρ′) (U u′) (ρ ρ′) v′ (ρ ρ′) w′ (ρ ρ′) 0. ∂t ¯ + + + ∂x ¯ + + ∂y ¯ + + ∂z ¯ + = Linearizing, we obtain 2 ∂ρ′ ∂ρ′ ∂ρ ρ0N w′ U v′ ¯ 0, (14.133) ∂t + ∂x + ∂y − g = 668 Geophysical Fluid Dynamics

2 1 where N gρ − (∂ρ/∂z). The perturbation density ρ can be written in terms of =− 0 ¯ ′ p′ by using the hydrostatic balance in equation (14.125), and subtracting the basic state (14.127). This gives

∂p′ 0 ρ′g, (14.134) =−∂z − which states that the perturbations are hydrostatic. Equation (14.133) then gives

1 ∂ ∂ ∂p′ dU ∂p′ w′ U , (14.135) =−ρ N 2 ∂t + ∂x ∂z − dz ∂x 0 !# $ " where we have written ∂ρ/∂y in terms of the thermal wind dU/dz. Using equa- tions (14.132) and (14.135),¯ the perturbation vorticity equation (14.130) becomes

2 2 ∂ ∂ 2 f ∂ p′ U p′ 0. (14.136) ∂t + ∂x ∇H + N 2 ∂z2 = # $! " This is the equation that governs the quasi-geostrophic perturbations on an eastward current U(z).

Wave Solution We assume that the flow is confined between two horizontal planes at z 0 and z H and that it is unbounded in x and y. Real flows are likely to be bounded= in the y =direction, especially in a laboratory situation of flow in an annular region, where the walls set boundary conditions parallel to the flow. The boundedness in y, however, simply sets up normal modes in the form sin(nπy/L), where L is the width of the channel. Each of these modes can be replaced by a periodicity in y. Accordingly, we assume wavelike solutions

i(kx ly ωt) p′ p(z) e + − . (14.137) =ˆ The perturbation vorticity equation (14.136) then gives

d2p ˆ α2p 0, (14.138) dz2 − ˆ = where N 2 α2 (k2 l2). (14.139) ≡ f 2 + The solution of equation (14.138) can be written as

H H p A cosh α z B sinh α z . (14.140) ˆ = − 2 + − 2 # $ # $ 17. Baroclinic Instability 669

Boundary conditions have to be imposed on solution (14.140) in order to derive an instability criterion.

Boundary Conditions The conditions are

w′ 0 at z 0,H. = =

The corresponding conditions on p′ can be found from equation (14.135) and U U0z/H . We obtain = ∂2p U z ∂2p U ∂p ′ 0 ′ 0 ′ 0 at z 0,H, −∂t∂z − H ∂x∂z + H ∂x = = where we have also used U U0z/H . The two boundary conditions are therefore = ∂2p U ∂p ′ 0 ′ 0 at z 0, ∂t∂z − H ∂x = = 2 2 ∂ p′ U0 ∂p′ ∂ p′ U0 0 at z H. ∂t∂z − H ∂x + ∂x∂z = =

Instability Criterion Using equations (14.137) and (14.140), the foregoing boundary conditions require

αH U αH A αc sinh 0 cosh 2 − H 2 ! " αH U αH B αc cosh 0 sinh 0, + − 2 + H 2 = ! " αH U0 αH A α(U0 c) sinh cosh − 2 − H 2 ! " αH U0 αH B α(U0 c) cosh sinh 0, + − 2 − H 2 = ! " where c ω/k is the eastward phase velocity. This= is a pair of homogeneous equations for the constants A and B. For nontrivial solutions to exist, the determinant of the coefficients must vanish. This gives, after some straightforward algebra, the phase velocity

U U αH αH αH αH c 0 0 tanh coth . (14.141) = 2 ± αH 2 − 2 2 − 2 &# $# $ Whether the solution grows with time depends on the sign of the radicand. The behavior of the functions under the radical sign is sketched in Figure 14.31. It is apparent that the first factor in the radicand is positive because αH/2 > tanh(αH/2) 670 Geophysical Fluid Dynamics

Figure 14.31 Baroclinic instability. The upper panel shows behavior of the functions in equation (14.141), and the lower panel shows growth rates of unstable waves.

for all values of αH. However, the second factor is negative for small values of αH for which αH/2 < coth(αH/2). In this range the roots of c are complex conjugates, with c U0/2 ici. Because we have assumed that the perturbations are of the form = ± exp( ikct), the existence of a nonzero ci implies the possibility of a perturbation − that grows as exp(kcit), and the solution is unstable. The marginal stability is given by the critical value of α satisfying

α H α H c coth c , 2 = 2 # $ whose solution is

αcH 2.4, = and the flow is unstable if αH < 2.4. Using the definition of α in equation (14.139), it follows that the flow is unstable if

HN 2.4 < . f √k2 l2 + 17. Baroclinic Instability 671

As all values of k and l are allowed, we can always find a value of k2 l2 low enough to satisfy the forementioned inequality. The flow is therefore always+ unstable (to low ). For a north–south wavenumber l 0, instability is ensured if the east–west wavenumber k is small enough such that= HN 2.4 < . (14.142) f k In a continuously stratified ocean, the speed of a long internal wave for the n 1 baroclinic mode is c NH/π, so that the corresponding internal Rossby radius= is c/f NH/πf . It is= usual to omit the factor π and define the Rossby radius in a continuously= stratified fluid as HN 0 . ≡ f The condition (14.142) for baroclinic instability is therefore that the east–west wave- length be large enough so that

λ>2.60.

However, the wavelength λ 2.60 does not grow at the fastest rate. It can be shown from equation (14.141) that= the wavelength with the largest growth rate is

λmax 3.90. =

This is therefore the wavelength that is observed when the instability develops. Typical values for f , N, and H suggest that λmax 4000 km in the atmosphere and 200 km in the ocean, which agree with observations.∼ Waves much smaller than the Rossby radius do not grow, and the ones much larger than the Rossby radius grow very slowly.

Energetics The foregoing analysis suggests that the existence of “weather waves” is due to the fact that small perturbations can grow spontaneously when superposed on an eastward current maintained by the sloping density surfaces (Figure 14.30). Although the basic current does have a vertical shear, the perturbations do not grow by extracting energy from the vertical shear field. Instead, they extract their energy from the potential energy stored in the system of sloping density surfaces. The energetics of the baroclinic instability is therefore quite different than that of the Kelvin–Helmholtz instability (which also has a vertical shear of the mean flow), where the perturbation Reynolds stress u′w′ interacts with the vertical shear and extracts energy from the mean shear flow. The baroclinic instability is not a shear flow instability; the Reynolds stresses are too small because of the small w in quasi-geostrophic large-scale flows. The energetics of the baroclinic instability can be understood by examining the equation for the perturbation kinetic energy. Such an equation can be derived by 672 Geophysical Fluid Dynamics

multiplying the equations for ∂u′/∂t and ∂v′/∂t by u′ and v′, respectively, adding the two, and integrating over the region of flow. Because of the assumed periodicity in x and y, the extent of the region of integration is chosen to be one wavelength in either direction. During this integration, the boundary conditions of zero normal flow on the walls and periodicity in x and y are used repeatedly. The procedure is similar to that for the derivation of equation (12.83) and is not repeated here. The result is dK g w′ρ′ dx dy dz, dt =− ) where K is the global perturbation kinetic energy

ρ0 2 2 K (u′ v′ )dxdydz. ≡ 2 + ) In unstable flows we must have dK/dt > 0, which requires that the volume inte- gral of w′ρ′ must be negative. Let us denote the volume average of w′ρ′ by w′ρ′. A negative w′ρ′ means that on the average the lighter fluid rises and the heavier fluid sinks. By such an interchange the center of gravity of the system, and therefore its potential energy, is lowered. The interesting point is that this cannot happen in a stably stratified system with horizontal density surfaces; in that case an exchange of fluid particles raises the potential energy. Moreover, a basic state with inclined density surfaces (Figure 14.30) cannot have w′ρ′ < 0 if the particle excursions are vertical. If, however, the particle excursions fall within the wedge formed by the constant den- sity lines and the horizontal (Figure 14.32), then an exchange of fluid particles takes lighter particles upward (and northward) and denser particles downward (and south- ward). Such an interchange would tend to make the density surfaces more horizontal, releasing potential energy from the mean density field with a consequent growth of the

Figure 14.32 Wedge of instability (shaded) in a baroclinic instability. The wedge is bounded by constant density lines and the horizontal. Unstable waves have a particle trajectory that falls within the wedge. 18. Geostrophic Turbulence 673 perturbation energy. This type of is called sloping convection. According to Figure 14.32 the exchange of fluid particles within the wedge of instability results in a net poleward transport of heat from the tropics, which serves to redistribute the larger solar heat received by the tropics. In summary, baroclinic instability draws energy from the potential energy of the mean density field. The resulting eddy motion has particle trajectories that are oriented at a small angle with the horizontal, so that the resulting heat transfer has a poleward component. The preferred scale of the disturbance is the Rossby radius.

18. Geostrophic Turbulence Two common modes of instability of a large-scale current system were presented in the preceding sections. When the flow is strong enough, such instabilities can make a flow chaotic or turbulent. A peculiarity of large-scale turbulence in the atmosphere or the ocean is that it is essentially two dimensional in nature. The existence of the Coriolis force, stratification, and small thickness of geophysical media severely restricts the vertical velocity in large-scale flows, which tend to be quasi-geostrophic, with the Coriolis force balancing the horizontal pressure gradient to the lowest order. Because vortex stretching, a key mechanism by which ordinary three-dimensional turbulent flows transfer energy from large to small scales, is absent in two-dimensional flow, one expects that the dynamics of geostrophic turbulence are likely to be fundamen- tally different from that of three-dimensional laboratory-scale turbulence discussed in Chapter 13. However, we can still call the motion “turbulent” because it is unpre- dictable and diffusive. A key result on the subject was discovered by the meteorologist Fjortoft (1953), and since then Kraichnan, Leith, Batchelor, and others have contributed to various aspects of the problem. A good discussion is given in Pedlosky (1987), to which the reader is referred for a fuller treatment. Here, we shall only point out a few important results. An important variable in the discussion of two-dimensional turbulence is enstro- phy, which is the mean square vorticity ζ 2. In an isotropic turbulent field we can define an energy spectrum S(K), a function of the magnitude of the wavenumber K, as

u2 ∞ S(K) dK. = )0 It can be shown that the enstrophy spectrum is K2S(K), that is,

ζ 2 ∞ K2S(K) dK, = )0 which makes sense because vorticity involves the spatial gradient of velocity. We consider a freely evolving turbulent field in which the shape of the velocity spectrum changes with time. The large scales are essentially inviscid, so that both 674 Geophysical Fluid Dynamics

energy and enstrophy are nearly conserved:

d ∞ S(K) dK 0, (14.143) dt = )0 d ∞ K2S(K) dK 0, (14.144) dt = )0 where terms proportional to the molecular viscosity ν have been neglected on the right-hand sides of the equations. The enstrophy conservation is unique to two-dimensional turbulence because of the absence of vortex stretching. Suppose that the energy spectrum initially contains all its energy at wavenumber K0. Nonlinear interactions transfer this energy to other wavenumbers, so that the sharp spectral peak smears out. For the sake of argument, suppose that all of the initial energy goes to two neighboring wavenumbers K1 and K2, with K1

S0 S1 S2, = + 2 2 2 K S0 K S1 K S2, 0 = 1 + 2

where Sn is the spectral energy at Kn. From this we can find the ratios of energy and enstrophy spectra before and after the transfer:

S1 K2 K0 K2 K0 − + , S2 = K0 K1 K1 K0 2 2 − 2 2+ (14.145) K S1 K K K 1 1 2 − 0 2 2 2 2 . K S2 = K K K 2 2 0 − 1 As an example, suppose that nonlinear smearing transfers energy to wavenum- bers K1 K0/2 and K2 2K0. Then equations (14.145) show that S1/S2 4 and 2 =2 1 = = K1 S1/K2 S2 4 , so that more energy goes to lower wavenumbers (large scales), whereas more= enstrophy goes to higher wavenumbers (smaller scales). This impor- tant result on two-dimensional turbulence was derived by Fjortoft (1953). Clearly, the constraint of enstrophy conservation in two-dimensional turbulence has prevented a symmetric spreading of the initial energy peak at K0. The unique character of two-dimensional turbulence is evident here. In small- scale three-dimensional turbulence studied in Chapter 13, the energy goes to smaller and smaller scales until it is dissipated by viscosity. In geostrophic turbulence, on the other hand, the energy goes to larger scales, where it is less susceptible to viscous dissipation. Numerical calculations are indeed in agreement with this behavior, which shows that the energy-containing eddies grow in size by coalescing. On the other hand, the vorticity becomes increasingly confined to thin shear layers on the eddy bound- aries; these shear layers contain very little energy. The backward (or inverse) energy cascade and forward enstrophy cascade are represented schematically in Figure 14.33. It is clear that there are two “inertial” regions in the spectrum of a two-dimensional turbulent flow, namely, the energy cascade region and the enstrophy cascade region. 18. Geostrophic Turbulence 675

Figure 14.33 Energy and enstrophy cascade in two-dimensional turbulence.

If energy is injected into the system at a rate ε, then the energy spectrum in the 2/3 5/3 energy cascade region has the form S(K) ε K− ; the argument is essentially the same as in the case of the Kolmogorov∝ spectrum in three-dimensional turbulence (Chapter 13, Section 9), except that the transfer is backwards. A dimensional argument also shows that the energy spectrum in the enstrophy cascade region is of the form 2/3 3 S(K) α K− , where α is the forward enstrophy flux to higher wavenumbers. There∝ is negligible energy flux in the enstrophy cascade region. As the eddies grow in size, they become increasingly immune to viscous dissipa- tion, and the inviscid assumption implied in equation (14.143) becomes increasingly applicable. (This would not be the case in three-dimensional turbulence in which the eddies continue to decrease in size until viscous effects drain energy out of the system.) In contrast, the corresponding assumption in the enstrophy conservation equation (14.144) becomes less and less valid as enstrophy goes to smaller scales, where viscous dissipation drains enstrophy out of the system. At later stages in the evolution, then, equation (14.144) may not be a good assumption. However, it can be shown (see Pedlosky, 1987) that the dissipation of enstrophy actually intensifies the process of energy transfer to larger scales, so that the red cascade (that is, transfer to larger scales) of energy is a general result of two-dimensional turbulence. The eddies, however, do not grow in size indefinitely. They become increasingly slower as their length scale l increases, while their velocity scale u remains constant. The slower dynamics makes them increasingly wavelike, and the eddies transform into Rossby-wave packets as their length scale becomes of order (Rhines, 1975)

u l (Rhines length), ∼ β . 676 Geophysical Fluid Dynamics

where β df/dy and u is the rms fluctuating speed. The Rossby-wave propagation results in= an anisotropic elongation of the eddies in the east–west (“zonal”) direction, while the eddy size in the north–south direction stops growing at √u/β. Finally, the velocity field consists of zonally directed jets whose north–south extent is of order √u/β. This has been suggested as an explanation for the existence of zonal jets in the atmosphere of the (Williams, 1979). The inverse energy cascade regime may not occur in the earth’s atmosphere and the ocean at midlatitudes because the Rhines length (about 1000 km in the atmosphere and 100 km in the ocean) is of the order of the internal Rossby radius, where the energy is injected by baroclinic instability. (For the inverse cascade to occur, √u/β needs to be larger than the scale at which energy is injected.) Eventually, however, the kinetic energy has to be dissipated by molecular effects at the Kolmogorov microscale η, which is of the order of a few millimeters in the ocean and the atmosphere. A fair hypothesis is that processes such as internal waves drain energy out of the mesoscale eddies, and breaking internal waves generate three-dimensional turbulence that finally cascades energy to molecular scales. A recent review of intense storm motion (lower atmosphere dynamics and ther- modynamics) was published by Chan (2005), whereas upper atmospheric motion was discussed by Haynes (2005). Oceanic flow transport was treated by Wiggins (2005).

Exercises 1. The Gulf Stream flows northward along the east coast of the United States with a surface current of average magnitude 2 m/s. If the flow is assumed to be in geostrophic balance, find the average slope of the sea surface across the current at a latitude of 45◦ N. [Answer: 2.1 cm per km] 6 2 2. A plate containing water (ν 10− m /s) above it rotates at a rate of 10 revolutions per minute. Find the depth= of the Ekman layer, assuming that the flow is laminar. 3. Assume that the atmospheric Ekman layer over the earth’s surface at a latitude 2 of 45◦ N can be approximated by an eddy viscosity of νv 10 m /s. If the geostrophic velocity above the Ekman layer is 10 m/s, what is the Ekman= transport across isobars? [Answer: 2203 m2/s] 4. Find the axis ratio of a hodograph plot for a semidiurnal in the middle of the ocean at a latitude of 45◦ N. Assume that the midocean tides are rotational surface gravity waves of long wavelength and are unaffected by the proximity of coastal boundaries. If the depth of the ocean is 4 km, find the wavelength, the phase velocity, and the group velocity. Note, however, that the wavelength is comparable to the width of the ocean, so that the neglect of coastal boundaries is not very realistic. 5. An internal Kelvin wave on the thermocline of the ocean propagates along the west coast of Australia. The thermocline has a depth of 50 m and has a nearly discontinuous density change of 2 kg/m3 across it. The layer below the thermocline Literature Cited 677

is deep. At a latitude of 30◦ S, find the direction and magnitude of the propagation speed and the decay scale perpendicular to the coast. 6. Using the dispersion relation m2 k2(N 2 ω2)/(ω2 f 2) for internal waves, show that the group velocity vector is= given by− −

(N 2 f 2)km cgx,cgz − m, k [ ]=(m2 k2)3/2(m2f 2 k2N 2)1/2 [ − ] + + [Hint: Differentiate the dispersion relation partially with respect to k and m.] Show that cg and c are perpendicular and have oppositely directed vertical components. Verify that cg is parallel to u.

7. Suppose the atmosphere at a latitude of 45◦ N is idealized by a uniformly stratified layer of height 10 km, across which the potential temperature increases by 50 ◦C. (i) What is the value of the buoyancy frequency N? (ii) Find the speed of a long gravity wave corresponding to the n 1 baroclinic mode. = (iii) For the n 1 mode, find the westward speed of nondispersive (i. e., very large = 1 wavelength) Rossby waves. [Answer: N 0.01279 s ; c1 40.71 m/s; = − = cx 3.12 m/s] =− 8. Consider a steady flow rotating between plane parallel boundaries a distance L apart. The angular velocity is ) and a small rectilinear velocity U is superposed. There is a protuberance of height h L in the flow. The Ekman and Rossby numbers are both small: Ro l, E l. Obtain≪ an integral of the relevant equations of motion that relates the modified≪ pressure≪ and the streamfunction for the motion, and show that the modified pressure is constant on streamlines.

Literature Cited

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Haynes, P. (2005). “Stratospheric dynamics.” Annual Review of Fluid Mechanics 37: 263–293. Holton, J. R. (1979). An Introduction to Dynamic , New York: Academic Press. Houghton, J. T. (1986). The Physics of the Atmosphere, London: Cambridge University Press. Kamenkovich, V. M. (1967). “On the coefficients of eddy diffusion and eddy viscosity in large-scale oceanic and atmospheric motions.” Izvestiya, Atmospheric and Oceanic Physics 3: 1326–1333. Kundu, P. K. (1977). “On the importance of friction in two typical continental : Off Oregon and Spanish Sahara,” in Bottom Turbulence, J. C. J. Nihoul, ed., Amsterdam: Elsevier. Kuo, H. L. (1949). “Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere.” Journal of Meteorology 6: 105–122. LeBlond, P. H. and L. A. Mysak (1978). Waves in the Ocean, Amsterdam: Elsevier. McCreary, J. P. (1985). “Modeling equatorial ocean circulation.” Annual Review of Fluid Mechanics 17: 359–409. Munk, W. (1981). “Internal waves and small-scale processes,” in Evolution of , B. A. Warren and C. Wunch, eds., Cambridge, MA: MIT Press. Pedlosky, J. (1971). “Geophysical fluid dynamics,” in Mathematical Problems in the Geophysical Sciences, W. H. Reid, ed., Providence, Rhode Island: American Mathematical Society. Pedlosky, J. (1987). Geophysical Fluid Dynamics, New York: Springer-Verlag. Phillips, O. M. (1977). The Dynamics of the Upper Ocean, London: Cambridge University Press. Prandtl, L. (1952). Essentials of Fluid Dynamics, New York: Hafner Publ. Co. Rhines, P. B. (1975). “Waves and turbulence on a β-plane.” Journal of Fluid Mechanics 69: 417–443. Taylor, G. I. (1915). “Eddy motion in the atmosphere.” Philosophical Transactions of the Royal Society of London A215: 1–26. Wiggins, S. (2005). “The dynamical systems approach to Lagrangian transport in oceanic flows.” Annual Review of Fluid Mechanics 37: 295–328. Williams, G. P. (1979). “Planetary circulations: 2. The Jovian quasi-geostrophic regime.” Journal of Atmo- spheric Sciences 36: 932–968.